9: Thermochemistry
Useful forms of energy are also available from a variety of chemical reactions other than combustion. For example, the energy produced by the batteries in a cell phone, car, or flashlight results from chemical reactions. This chapter introduces many of the basic ideas necessary to explore the relationships between chemical changes and energy, with a focus on thermal energy.
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- 9.1: Introduction
- Useful forms of energy are also available from a variety of chemical reactions other than combustion. For example, the energy produced by the batteries in a cell phone, car, or flashlight results from chemical reactions. This chapter introduces many of the basic ideas necessary to explore the relationships between chemical changes and energy, with a focus on thermal energy.
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- 9.2: Energy Basics
- Energy is the capacity to do work (applying a force to move matter). Heat is energy that is transferred between objects at different temperatures; it flows from a high to a low temperature. Chemical and physical processes can absorb heat (endothermic) or release heat (exothermic). The SI unit of energy, heat, and work is the joule (J). Specific heat and heat capacity are measures of the energy needed to change the temperature of a substance or object.
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- 9.3: Calorimetry
- Calorimetry is used to measure the amount of thermal energy transferred in a chemical or physical process. This requires careful measurement of the temperature change that occurs during the process and the masses of the system and surroundings. These measured quantities are then used to compute the amount of heat produced or consumed in the process using known mathematical relations. Calorimeters are designed to minimize energy exchange between the system and its surroundings.
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- 9.4: Enthalpy
- If a chemical change is carried out at constant pressure and the only work done is caused by expansion or contraction, q for the change is called the enthalpy change with the symbol ΔH. Examples of enthalpy changes include enthalpy of combustion, enthalpy of fusion, enthalpy of vaporization, and standard enthalpy of formation. If the enthalpies of formation are available for the reactants and products of a reaction, the enthalpy change can be calculated using Hess’s law.


11.6: Notes
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- Jade Miller
- University of California Press
1 Amelia Arsenault and Manuel Castells, “The Structure and Dynamics of Global Multi-media Business Networks,” International Journal of Communication 2 (2008): 707–748.
2 Toby Miller, Nitin Govil, John McMurria, Richard Maxwell, and Ting Wang, Global Hollywood 2 (London: British Film Institute, 2008).
3 This chapter does not directly compare Nollywood to Hollywood. To compare the two is to suggest that Nollywood is a less-funded imitator of Hollywood, when it is more accurate to view Nollywood as on its own trajectory, growing out of and flourishing in a different position in the global economy. However, in the context of this edited volume, I engage in a general comparison of Nollywood to more formal global media industries worldwide, as part of this book’s charge to illuminate how global concerns about labor issues play out across a diversity of contexts.
4 It has become popular in recent years to cite a UNESCO statistic that Nollywood is second only to Bollywood in number of titles produced per year, but scholars have questioned the significance of such a statistic in comparing theatrical and home-viewing based industries. See Carmela Garritano, “Introduction: Nollywood—an Archive of African Worldliness,” Black Camera 5.2 (2013): 44–52; and other articles in that Nollywood special issue of Black Camera .
5 See Brian Larkin, “Degraded Images, Distorted Sounds: Nigerian Video and the Infrastructure of Piracy,” Public Culture 16.2 (2004): 289–314, for a full argument on informal distribution as a source of industrial strength.
6 Despite recent efforts in formalization in cinema, online, and satellite distribution, the core of industry profits are still in physical copies sold in networks of domestic open-air markets.
7 Jade Miller, “Global Nollywood: The Nigerian Movie Industry and Alternative Global Networks in Production and Distribution,” Global Media and Communication 8 (2012): 117–133.
8 See ibid. for an analysis of Nollywood’s formal inputs, such as cameras and sound equipment.
9 See Ramon Lobato, “Creative Industries and Informal Economies: Lessons from Nollywood,” International Journal of Cultural Studies 13 (2010): 337–354.
10 Matthew Gandy, “Learning from Lagos,” New Left Review 33 (2005): 36–52; Alessandro Jedlowski, “Nigerian Videos in the Global Arena: The Postcolonial Exotic Revisited,” Global South 7.1 (2013): 157–178; Nyasha Mboti, “Nollywood’s Aporias Part 1: Gatemen,” Journal of African Cinemas 6 (2014): 49–70.
11 Jonathan Haynes, “Nollywood in Lagos, Lagos in Nollywood Films,” Africa Today 54.2 (2007): 131–150.
12 Figures, to be taken with many grains of salt, are derived from Rem Koolhaas, Harvard Project on the City, Stefano Boeri, Sanford Kwinter, Nadia Tazi, and Hans Ulrich Obrist, Mutations (New York: ACTAR, 2000).
13 Ibid.
14 Koolhaas et al., Mutations . And counting comparative global generator density by neighborhood is perhaps the epitome of figures that could never be accurately counted.
15 See Elizabeth Rosenthal, “Nigeria Tested by Rapid Rise in Population,” New York Times , April 14, 2012, www.nytimes.com/2012/04/15/world/africa/in-nigeria-a-preview-of-an-overcrowded-planet. html. This statistic is at the high end of estimates; the population may well be lower.
16 National Bureau of Statistics of Nigeria, Annual Abstract of Statistics, 2012 , 115–117, www.nigerianstat.gov.ng/nbslibrary/nbs-annual-abstract-of-statistics/nbs-annual-abstract-of-statistics.
17 Those who made the movie see no direct profits from this.
18 Jonathan Haynes, “New Nollywood: Kunle Afolayan,” Black Camer a 5.2 (2013): 53–73; Moradewun Adejunmobi, “Evolving Nollywood Templates for Minor Transnational Film,” Black Camera 5.2 (2014): 74–94.
19 Connor Ryan, “Nollywood and the Limits of Informality: A Conversation with Tunde Kelani, Bond Emeruwa, and Emem Isong,” Black Camera 5.2 (2013): 168–185.
20 See Larkin, “Degraded Images, Distorted Sounds.”
21 Ibid.
22 Those advocating loudest for more formality are those who would likely lead the industry were the marketers to lose control. This small group of big-name Nollywood producers and directors make movies that emerge from a largely separate self-financed system known sometimes as “New Nollywood.” This chapter, however, deals not with “New Nollywood” but with the bulk of the industry, which produces the majority of titles and employs the vast majority of workers.
23 It shouldn’t be all that surprising that the institution of guilds has proven so popular in structuring primarily informal Nollywood. Nigerian society is full of organizations and leadership positions. It is not uncommon for people to spend their little spare time going from meeting to meeting: church governance groups, church committees, groups of those originally from the same village, and so on. It seems that anyone who is anyone (and many who are, in effect, nobodies) holds or has held a leadership position in some organization or another. Ascendency to leadership in any organization is afforded a high degree of respect and importance, and leaders of even the smallest of these organizations are usually hailed by their title in public.
24 Richard Caves, Creative Industries: Contracts between Art and Commerce (Cambridge, MA: Harvard University Press, 2002).
25 Annalee Saxenian, Regional Advantage: Culture and Competition in Silicon Valley and Route 128 (Cambridge, MA: Harvard University Press, 1996).
26 Ramon Lobato, Shadow Economies of Cinema: Mapping Informal Film Distribution (London: British Film Institute, 2012).
27 A popular cinematographer suggests that ten to fifteen days is the bare minimum for a movie shoot, while a “good” movie will take twenty-one or more days. Shooting has been taking longer in recent years.
28 Caves, Creative Industries .
29 Elizabeth Currid, The Warhol Economy: How Fashion, Art, and Music Drive New York City (Princeton, NJ: Princeton University Press, 2007).
30 In Nigeria, a wealthy, successful man is usually referred to as “big,” and underlings and servants are often referred to as “boys,” no matter their age.
31 With the exception of sudden dramatic overtures, such as former president Goodluck Jonathan’s disjointed multimillion-dollar funding scheme late in his tenure or the censorship board’s ill-fated efforts to restructure distribution.


Early childhood is a time when children are especially attracted to motion and song.  Even the booth at a restaurant affords the opportunity to slide around in the seat or disappear underneath and imagine being a sea creature in a cave! Of course, this can be frustrating to a caregiver, but it’s the business of early childhood.
Gross Motor Skills
Children continue to improve their gross motor skills as they run and jump. They frequently ask their caregivers to “look at me” while they hop or roll down a hill. Children’s songs are often accompanied by arm and leg movements or cues to turn around or move from left to right.
Gross Motor Milestones
Here is a table showing the progression of gross motor skills that children will typically develop during early childhood:
Here are some activities focused on play that young children enjoy and that support their gross motor skill development.
Tricycle
Slides
Swings
Sit-n-Spin
Mini trampoline
Bowling pins (can use plastic soda bottles also)
Tent (try throwing blankets over chairs and other furniture to make a fort)
Playground ladders
Suspension bridge on playground
Tunnels (try throwing a bean bag chair underneath for greater challenge)
Ball play (kick, throw, catch)
Simon Says
Target games with bean bags, ball, etc.
Dancing/moving to music
Pushing self on scooter or skateboard while on stomach
Fine Motor Skills
Fine motor skills are also being refined as they continue to develop more dexterity, strength, and endurance. Fine motor skills are very important as they are foundational to self-help skills and later academic abilities (such as writing).
Fine Motor Milestones
Here is a table showing how fine motor skills progress during early childhood for children that are typically developing.
Here are some fun activities that will help children continue to refine their fine motor abilities. Fine motor skills are slower to develop than gross motor skills, so it is important to have age appropriate expectations and play-based activities for children.
Pouring water into a container
Drawing and coloring
Using scissors
Finger painting
Fingerplays and songs (such as the Itsy, Bitsy Spider)
Play dough
Lacing and beading
Practicing with large tweezers, tongs, and eye droppers


5.4: Hyphens and Dashes
- Demonstrate the proper use of hyphens and dashes
Hyphens
The Oxford Manual of Style once stated, “If you take hyphens seriously, you will surely go mad.” Hyphens belong to that category of punctuation marks that will hurt your brain if you think about them too hard, and, like commas, people disagree about their use in certain situations. Nevertheless, you will have to use them regularly because of the nature of academic and professional writing. If you learn to use hyphens properly, they help you to write efficiently and concretely.
The Hyphen’s Function
Fundamentally, the hyphen is a joiner. It can join several different types of things:
- two nouns to make one complete word (kilogram-meter)
- an adjective and a noun to make a compound word (accident-prone)
- two words that, when linked, describe a noun (agreed-upon sum, two-dimensional object)
- a prefix with a noun (un-American)
- double numbers (twenty-four)
- numbers and units describing a noun (1000-foot face; a 10-meter difference)
- “self” words (self-employed, self-esteem)
- new word blends (cancer-causing, cost-effective)
- prefixes and suffixes to words, in particular when the writer wants to avoid doubling a vowel or tripling a consonant (anti-inflammatory; shell-like)
- multiple adjectives with the same noun (blue- and yellow-green beads; four- and five-year-olds)
A rule of thumb for the hyphen is that the resulting word must act as one unit; therefore, the hyphen creates a new word that has a single meaning. Usually, you can tell whether a hyphen is necessary by applying common sense and mentally excluding one of the words in question, testing how the words would work together without the hyphen. For example, the phrases “high-pressure system,” “water-repellent surface,” and “fuel-efficient car” would not make sense without hyphens, because you would not refer to a “high system,” a “water surface,” or a “fuel car.” As your ears and eyes become attuned to proper hyphenation practices, you will recognize that both meaning and convention dictate where hyphens fit best.
Examples of Properly Used Hyphens
Some examples of properly used hyphens follow. Note how the hyphenated word acts as a single unit carrying a meaning that the words being joined would not have individually.
| small-scale study | two-prong plug | strength-to-weight ratio | high-velocity flow | frost-free lawn |
| self-employed worker | one-third majority | coarse-grained wood | decision-making process | blue-green algae |
| air-ice interface | silver-stained cells | protein-calorie malnutrition | membrane-bound vesicles | phase-contrast microscope |
| long-term-payment loan | cost-effective program | time-dependent variable | radiation-sensitive sample | long-chain fatty acid |
When Hyphens Are Not Needed
By convention, hyphens are not used after words ending in – ly , nor when the words are so commonly used in combination that no ambiguity results. In these examples, no hyphens are needed:
| finely tuned engine | blood pressure | sea level |
| real estate | census taker | atomic energy |
| civil rights law | public utility plant | carbon dioxide |
Note: Phrases like containing the word well like well known are contested. Well is an adverb, and thus many fall into the school of thought that a hyphen is unnecessary. However, others say that leaving out the hyphen may cause confusion and therefore include it ( well-known ). The standard in MLA is as follows: When it appears before the noun, well known should be hyphenated. When it follows the noun, no hyphenation is needed.
- She is a well-known person.
- She is well known .
Prefixes and Suffixes
Most prefixes do not need to be hyphenated; they are simply added in front of a noun, with no spaces and no joining punctuation necessary. The following is a list of common prefixes that do not require hyphenation when added to a noun:
| after | anti | bi | bio | co |
| cyber | di | down | hetero | homo |
| infra | inter | macro | micro | mini |
| non | photo | poly | stereo | thermo |
When prefixes are added to a proper noun, they require a hyphen (e.g., nonviolent , but non-European ).
Note: The prefix re generally doesn’t require a hyphen. However, when leaving out a hyphen will cause confusion, one should be added. Look at the following word pairs, for example:
- resign (leave a position) v. re-sign (sign the paper again)
- recreation (an activity of leisure) v. re-creation (create something again)
Common suffixes also do not require hyphenation, assuming no ambiguities of spelling or pronunciation arise. Typically, you do not need to hyphenate words ending in the following suffixes:
| able | less | fold | like | wise |
Commonly Used Word Blends
Also, especially in technical fields, some words commonly used in succession become joined into one. The resulting word’s meaning is readily understood by technical readers, and no hyphen is necessary. Here are some examples of such word blends, typically written as single words:
| blackbody | groundwater | airship |
| downdraft | longwall | upload |
| setup | runoff | blowout |
Dashes
The dash functions almost as a colon does in that it adds to the preceding material, but with extra emphasis. Like a caesura (a timely pause) in music, a dash indicates a strong pause, then gives emphasis to material following the pause. In effect, a dash allows you to redefine what was just written, making it more explicit. You can also use a pair of dashes in place of parentheses, to frame an interruptive or parenthetical-type comment that you do not want to de-emphasize.
- Jill Emery confirms that Muslim populations have typically been ruled by non-Muslims—specifically Americans, Russians, Israelis, and the French.
- The dissolution took 20 minutes—much longer than anticipated—but measurements were begun as soon as the process was completed.
There is no “dash” button on a computer keyboard. Instead, create it by typing the hyphen button twice in a row; or use the “symbol” option in your word processor; or use the Mac shortcut option + shift + —.
The Em-dash
The dash we typically use is technically called the “em dash,” which gets its name from back in the days of typesetting on old-fashioned printing presses. It is a dash that is the width of an M , so it is significantly longer than the hyphen.
The En-dash
There is also an “en dash”—whose length is that of an N , between that of the hyphen and the em dash, and its best usage is to indicate inclusive dates and numbers:
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July 6–September 17
- The date range began on July 6 and ended on September 17.
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Barack Obama (1961–)
- This indicates the year a person was born, as well as the fact that he or she is still alive.
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pp. 148–56
- This indicates pages 148 through 156. With number ranges, you can remove the first digit of the second number if it’s the same as the first number’s.
The en dash can also be used for flight or train routes.
- The London–Paris train will be running thirty minutes late today.
Like the em dash, the en dash is not on the standard computer keyboard. Select it from word processor’s symbol map (or if you have a Mac, you can type option + – ), or it may even be inserted automatically by your word processor when you type inclusive numbers or dates with a hyphen between them. In most contexts, a hyphen can serve as an en dash, but in professional publications—especially in the humanities—an en dash is correct.
When you type the hyphen, en dash, and em dash, no spaces should appear on either side of the punctuation mark.
https://assessments.lumenlearning.co...essments/20236
https://assessments.lumenlearning.co...essments/20237
https://assessments.lumenlearning.co...essments/20238
Click through this presentation and answer the questions to test your understanding of when to use dashes and hyphens.
https://lumenlearning.h5p.com/content/1290909726859410368/embed
Contributors and Attributions
- Revision and adaptation. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Original Icons. Provided by : Lumen Learning. License : CC BY: Attribution
- Style For Students Online. Authored by : Joe Schall. Provided by : The Pennsylvania State University. Located at : https://www.e-education.psu.edu/styleforstudents/ . Project : Penn State's College of Earth and Mineral Sciences' OER Initiative. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Modification of John Cage (errors added). Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/John_Cage . License : CC BY-SA: Attribution-ShareAlike
- Hyphens and Dashes Activity. Authored by : Excelsior OWL. Located at : https://owl.excelsior.edu/writing-refresher/grammar-refresher/dashes-hyphens/ . License : CC BY: Attribution


2.11: How to Read Effectively in Various Disciplines
- Explain tips for reading effectively in various disciplines (sciences, social sciences, and math)
Reading Online
It’s more and more likely that you’ll have to do a large portion of your reading for college online
(like this!). Because it is sometimes harder to focus on online readings when there are so many distractions readily available at your fingertips, it’s important to be extra thoughtful as you work through online text. Just as you would with a traditional textbook, take the time to minimize distractions. Use programs such as Mercury Reader, task-trackers, timers, and blockers to reduce unwanted distractions. Preview the text by scrolling through a section, making note of headers and keywords. Be careful when clicking on hyperlinks that you don’t end up losing sight of your initial reading. Navigate through the material in an intentional way. Take notes and annotate or outline either on your device, or with a pen and paper. These strategies will assist you with reading, no matter the subject area.
Reading in the Sciences
Reading in the sciences is likely different than reading in the humanities or for some of your other classes.
Explore the Textbook and Assignments
In a science textbook, it’s helpful to develop a base understanding of how the material is organized. Go over the course outline and the table of contents and compare the two. In addition, explore the lab manual, if you have one.
You also benefit from knowing what you’re required to learn. Read the introduction of your assigned chapter and connect it with previous chapters or your prior knowledge of the topic. It is important to read the headings, subheadings, summary, and review questions. Remember that most science texts contain review questions; use them to guide your readings.
Look over the pictures, tables, diagrams, photographs, and other images. Sometimes those elements are easier to understand than the words.
In the sciences, it is especially helpful to take notes, and learn basic terminology and vocabulary. Studying scientific root prefixes and suffixes and using context clues can help you understand new words.
Analyze For Comprehension
Scientific texts usually follow the same writing patterns. Once you can recognize and analyze them, your comprehension will increase.
The Classification Pattern: used by scientists to group and sub-group various things, objects, or areas. For example, a scientist who wishes to discuss the structure of a plant may break his topic into various subheadings as roots, stems, leaves, or flowers. Recognizing these structural parts in order of importance or position is essential to good comprehension and note-taking.
The Process Description Pattern: what the process is and how the process works. You need to understand what the description pattern is about. Is it about the process? or how the process works?
The Factual-Statement Pattern: facts are usually used in defining things, in comparing or contrasting things, and in citing examples or illustrations. In science, the word “fact” has a more exacting meaning than in other areas.
The Problem-Solving Pattern: usually found in passages from science texts which describe or recount past scientific problems, or scientific discoveries made through experimentation. When you’re confronted with the problem solving pattern, use the following questions to help you understand and analyze the passages.
- What is the question or problem?
- How was the question answered?
- How do we know it was answered?
In addition, application of these questions can help you to separate the major and minor points.
Experiment-Instruction Pattern: to understand this pattern and to make sure that you follow the instructions exactly, use the following questions.
- What is the purpose of the experiment?
- What equipment is needed?
- What, in order, are the basic steps involved?
- What are the results?
Usually you must alternate between the reading matter and the experimental tool, so have the questions firmly in mind before attempting the experiment. In addition, use the questions when you have been given an assignment from your lab manual.
The Combination Pattern: not all science texts follow one pattern. Sometimes the writer may use a combination of patterns. For instance, a reading passage may begin with factual statement of definition, move to classifying the components or parts of the term being classified, and end up discussing a process. An awareness of all patterns is needed in this case to aid in distinguishing the main ideas and supporting details in the various pattern used.
Reading in the Social Sciences
Faced with a long list of readings in any social science field, you need to learn to read extensively as well as intensively; it is rarely practical to read everything word for word and line for line. Although close textual reading and interpretation is part of social science tradition, it is often not possible, especially for introductory and intermediate level survey courses. Instead of trying to read every line and word, consider the following suggestions for more efficient and effective course reading.
Organize reading over the weeks and months
Look over the material to be covered (syllabus and tables of contents in assigned books). Estimate the amount of reading for the semester and try to divide the work on a weekly basis. Try to keep your work and pace steady. It will become less burdensome and easier to manage.
Begin any reading assignment by reading the abstract, preface, introductions, and conclusions
The beginning and end are frequently the most important parts of any text because the author often signals his or her major themes and arguments. It is necessary, however, to look over, sometimes very carefully and completely, the central portions of the text to identify the evidence provided for the major themes/theses. Often, the topic (first) sentences of paragraphs provide the links in the author’s argument.
Mechanics of reading and note-taking
Read the text and make marginal notes indicating what seemed like the strongest parts of the text. When you have completed a once-through of the text, go back and take notes in outline form, by paraphrasing sentences or paragraphs until you have reduced the many pages of text to a few pages. (Make sure to keep an accurate citation to the work so that any future use of these notes and paraphrases can be appropriately cited—you do not want to find yourself engaged in plagiarism.)
- Classify the book or article according to kind and subject matter. Into what genre does that work fit? What is the book about?
- Number the major parts in their order and relations. Outline these as you have outlined the whole.
- Define the specific problem or problems the author has tried to solve. What question does the author claim to address? You might also want to think about how this reading fits into the course. Why did the instructor place the reading at this point in the course? What is the topic on the syllabus? How does this reading provide an answer or information for this topic?
- What theoretical statements does the author make? A theoretical statement proposes a relationship. For example, structural theories of deviance suggest that deviance (that which is to be explained) is a consequence of the structure (organization of the parts) of a society. In other words, social structure produces deviance.
- What are the concepts and variables used? Become familiar with the author by defining keywords. Know the details of the argument. In the example above: what is social structure? What is meant by deviance? Do structural theorists/ writers assume the reader knows what is meant by social structure? Do you need to find out what this means in order to understand the reading?
- How does the author’s argument/ position compare with that of others who address the same question or related questions? Where are the points of similarity and difference?
- What value judgments does the author make? What values does the author assume readers will share? What assumptions does the author make that may be contestable?
- What is the author’s methodology? (Here you should be concerned not only with the methods used but the kinds of arguments implied or given about what methods are more or less appropriate.) What constitutes evidence in this reading? Know the author’s arguments by finding them in, or constructing them out of, sequences of sentences.
- Determine which of the problems the author has solved and which she has not. Of those not solved, decide which the author knows she has failed to solve. If you disagree with the author, on what basis do you rest your disagreement? Is the author uninformed, misinformed, illogical, imprecise, or incomplete? Criticize fairly; do not pass judgment based on personal opinion, taste, or preference. Is the argument internally consistent? Does the evidence (both that presented by the author and other evidence in the field) support the argument?
Reading for Math Class
Unlike some classes where you are able to skim over some readings to look for specific information, readings for a math class require careful reading of every word. Math textbooks rely heavily on practice problems, so you will typically find a few paragraphs of explanation and introduction, then problems. You want to make sure you have a pencil and paper in hand, and even if you think you know how to do the problem, it’s a good idea to work through every example. Be sure to plan enough time for this, and work slowly through each step. Write down your notes during each step as well, so if you are confused about anything, you’ll be able to pinpoint which step is giving you a hard time. Here are some tips for working through a solved problem in the textbook:
- Work through the problem one step at a time
- Close the book and try to work it again on your own
- Repeat until you can reproduce the solution with the book closed
- Try not to memorize the solution
- Keep track of “what to do” to move from each line to the next
- It’s okay if your version has more lines than the author’s (it may take you two or three steps to accomplish what the author does in one). This is a good sign that you’re thinking for yourself!
- Be patient. It’s common to spend an hour or two on a single page. It’s worth your time to learn the process.
After you can work through the solved problems on your own, the homework should be much easier since there will be similar problems. Time spent on problems the author has solved for you will pay off in higher grades.
Don’t worry if the author or your instructor makes it “look easy.” They work hard when you aren’t looking. The author chooses “cleaned up” problems for his or her textbook. Teachers do the same thing when preparing for a lecture. Good math is messy! If you get really lost, scrap your work and start over on that problem.
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Image of scientist holding molecule. Authored by : Philip Bond. Located at : https://flic.kr/p/NZQuw . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- How to Read Effectively in the Sciences. Authored by : John Croom III. Located at : http://croomphysics.com/index.php?p=reading_science&lb=reading . Project : Croom Physics. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Image of colored tubes. Authored by : poisson lucas. Located at : https://flic.kr/p/eUJs3s . License : CC BY-NC: Attribution-NonCommercial
- Guide to Reading Social Science: How to work through long reading assignments. Authored by : Susan S. Sibley. Provided by : MIT. Located at : http://ocw.mit.edu/courses/anthropology/21a-245j-power-interpersonal-organizational-and-global-dimensions-fall-2005/study-materials/guidereading.pdf . Project : MIT Open Courseware. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Image of math problems. Authored by : Jeremy Mikkola. Located at : https://flic.kr/p/379T7b . License : CC BY: Attribution
- How to Read a Math Textbook. Provided by : Great Basin College. Located at : http://www.gbcnv.edu/documents/ASC/docs/00000075.pdf . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives


Page 6.9: Arcs, Angles, and Trig Function Values
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Learning Objectives
After studying this section, we should understand the concepts motivated by the questions below and be able to write precise, coherent answers to these questions:
- How do we measure angles using degrees?
- What do we mean by the radian measure of an angle? How is the radian measure of an angle related to the length of an arc on the unit circle?
- Why is radian measure important?
- How do we convert from radians to degrees and from degrees to radians?
- How do we determine the values of both \(x\) and \(y\) when parameters on the domain are restricted?
- What trigonometric functions are positive in the first, second, third and fourth quadrants?
- How do we use one trigonometric function to represent or describe another?
The ancient civilization known as Babylonia was a cultural region based in southern Mesopotamia, which is present-day Iraq. Babylonia emerged as an independent state around 1894 BCE. The Babylonians developed a system of mathematics that was based on a sexigesimal (base 60) number system. This was the origin of the modern day usage of 60 minutes in an hour, 60 seconds in a minute, and 360 degrees in a circle.
Many historians now believe that for the ancient Babylonians, the year consisted of 360 days, which is not a bad approximation given the crudeness of the ancient astronomical tools. As a consequence, they divided the circle into 360 equal length arcs, which gave them a unit angle that was 1/360 of a circle or what we now know as a degree. Even though there are 365.24 days in a year, the Babylonian unit angle is still used as the basis for measuring angles in a circle. Figure \(\PageIndex{1}\) shows a circle divided up into 6 angles of 60 degrees each, which is also something that fit nicely with the Babylonian base-60 number system.
Figure \(\PageIndex{1}\): A circle with six 60-degree angles.
We often denote a line that is drawn through 2 points A and B by \(\overleftrightarrow{AB}\). The portion of the line \(\overleftrightarrow{AB}\) that starts at the point A and continues indefinitely in the direction point of point B is called ray AB and is denoted by \(\overrightarrow{AB}\). The point A is the initial point of ray \(\overrightarrow{AB}\). An angle is formed by rotating a ray about its endpoint. The ray in its initial position is called the initial side of the angle, and the position of the ray after it has been rotated is called the terminal side of the ray. The endpoint of the ray is called the vertex of the angle.
Figure \(\PageIndex{2}\): An angle including some notation.
Figure \(\PageIndex{2}\) shows the ray \(\overrightarrow{AB}\) rotated about the point A to form an angle. The terminal side of the angle is the ray \(\overrightarrow{AC}\). We often refer to this as angle BAC, which is abbreviated as \(\angle{BAC}\). We can also refer to this angle as angle \(CAB\) or \(\angle{CAB}\). If we want to use a single letter for this angle, we often use a Greek letter such as \(\alpha\)(alpha). We then just say the angle ̨. Other Greek letters that are often used are \(\beta\)(beta), \(\gamma\)(gamma), \(\theta\)(theta), \(\phi\)(phi), and \(\rho\)(rho).
Arcs and Angles
To define the trigonometric functions in terms of angles, we will make a simple connection between angles and arcs by using the standard position of an angle. When the vertex of an angle is at the origin in the \(xy\)-plane and the initial side lies along the positive x-axis, we see that the angle is in standard position . The terminal side of the angle is then in one of the four quadrants or lies along one of the axes. When the terminal side is in one of the four quadrants, the terminal side determines the so-called quadrant designation of the angle. See Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): Standard position of an angle in the second quadrant.
Exercise \(\PageIndex{1}\)
Draw an angle in standard position in
- the first quadrant;
- the third quadrant; and
- the fourth quadrant.
- Answer
-
These graphs show positive angles in standard position. The one on the left has its terminal point in the first quadrant, the one in the middle has its terminal point in the third quadrant, and the one on the right has its terminal point in the fourth quadrant.
If an angle is in standard position, then the point where the terminal side of the angle intersects the unit circle marks the terminal point of an arc as shown in Figure 1.11. Similarly, the terminal point of an arc on the unit circle determines a ray through the origin and that point, which in turn defines an angle in standard position. In this case we say that the angle is subtended by the arc. So there is a natural correspondence between arcs on the unit circle and angles in standard position. Because of this correspondence, we can also define the trigonometric terminal side functions in terms of angles as well as arcs. Before we do this, however, we need to discuss two different ways to measure angles.
Figure \(\PageIndex{4}\): An arc and its corresponding angle.
Degrees Versus Radians - A Review (Because we ALL need one!)
There are two ways we will measure angles – in degrees and radians. When we measure the length of an arc, the measurement has a dimension (the length, be it inches, centimeters, or something else). As mentioned in the introduction, the Babylonians divided the circle into 360 regions.  The unit measure of \(1^\circ\) is an angle that is 1/360 of the central angle of a circle. Figure \(\PageIndex{1}\) shows 6 angles of \(60^\circ\) each. The degree \(^\circ\) is a dimension, just like a length. So to compare an angle measured in degrees to an arc measured with some kind of length, we need to connect the dimensions. We can do that with the radian measure of an angle.
Radians will be useful in that a radian is a dimensionless measurement. We want to connect angle measurements to arc measurements, and to do so we will directly define an angle of 1 radian to be an angle subtended by an arc of length 1 (the length of the radius) on the unit circle as shown in Figure \(\PageIndex{5}\).
Figure \(\PageIndex{5}\): One radian.
Definition: Radian
An angle of one radian is the angle in standard position on the unit circle that is subtended by an arc of length 1 (in the positive direction).
This directly connects angles measured in radians to arcs in that we associate a real number with both the arc and the angle. So an angle of 2 radians cuts off an arc of length 2 on the unit circle, an angle of 3 radians cuts of an arc of length 3 on the unit circle, and so on. Figure 1.13 shows the terminal sides of angles with measures of 0 radians, 1 radian, 2 radians, 3 radians, 4 radians, 5 radians, and 6 radians. Notice that \(2\pi \approx 6.2832\) and so \(6 < 2\pi\) as shown in Figure \(\PageIndex{6}\).
Figure \(\PageIndex{6}\): Angles with Radian Measure 1, 2, 3, 4, 5, and 6
We can also have angles whose radian measure is negative just like we have arcs with a negative length. The idea is simply to measure in the negative (clockwise) direction around the unit circle. So an angle whose measure is \(-1\) radian is the angle in standard position on the unit circle that is subtended by an arc of length 1 in the negative (clockwise) direction.
So in general, an angle (in standard position) of \(t\) radians will correspond to an arc of length \(t\) on the unit circle. This allows us to discuss the sine and cosine of an angle measured in radians. That is, when we think of sin\((t)\) and cos\((t)\), we can consider \(t\) to be:
- a real number;
- the length of an arc with initial point \((1, 0)\) on the unit circle;
- the radian measure of an angle in standard position.
When we draw a picture of an angle in standard position, we often draw a small arc near the vertex from the initial side to the terminal side as shown in Figure \(\PageIndex{7}\), which shows an angle whose measure is \(\dfrac{3}{4}\pi\) radians.
Example \(\PageIndex{1}\)
1. Draw a terminal ray in the quadrant defined by the given \(\theta\) restrictions; measuring from standard position with a radian measure:
Let \(\sin({\theta}\)
- \(\dfrac{\pi}{2}\)
- \(\pi\) radians.
- \(\dfrac{3\pi}{2}\) radians.
- \(\dfrac{-3\pi}{2}\) radians.
2. What is the degree measure of each of the angles in part (1)?
- Answer
-
- \[90^\circ\]
- \[180^\circ\]
- \[270^\circ\]
- \[-270^\circ\]
Conversion Between Radians and Degrees
Radian measure is the preferred measure of angles in mathematics for many reasons, the main one being that a radian has no dimensions. However, to effectively use radians, we will want to be able to convert angle measurements between radians and degrees.
Recall that one wrap of the unit circle corresponds to an arc of length \(2\pi\), and an arc of length \(2\pi\) on the unit circle corresponds to an angle of \(2\pi\) radians. An angle of \(360^\circ\) is also an angle that wraps once around the unit circle, so an angle of \(360^\circ\) is equivalent to an angle of \(2\pi\) radians, or
- each degree is \(\dfrac{\pi}{180}\) radians,
- each radian is \(\dfrac{180}{\pi}\) degrees.
Notice that 1 radian is then \(\dfrac{180}{\pi} \approx 57.3^\circ\), so a radian is quite large compared to a degree. These relationships allow us to quickly convert between degrees and radians.
Example \(\PageIndex{1}\)
- If an angle has a degree measure of 35 degrees, then its radian measure can be calculated as follows:
\[35 \space degrees \times \dfrac{\pi \space radians}{180 \space degrees} = \dfrac{35\pi}{180} \space radians\]
Rewriting this fraction, we see that an angle with a measure of 35 degrees has a radian measure of \(\dfrac{7\pi}{36}\) radians.
- If an angle has a radian measure of \(\dfrac{3\pi}{10}\) radians, then its degree measure can be calculated as follows:
\[\dfrac{3\pi}{10} \space radians \space \times \dfrac{180 \space degrees}{\pi \space radians}= \dfrac{540}{10} \space degrees\]
So an angle with a radian measure of \(\dfrac{3\pi}{10}\) has an angle measure of 54\(^\circ\).
IMPORTANT NOTE
Since a degree is a dimension, we MUST include the degree mark \(^\circ\) whenever we write the degree measure of an angle. A radian has no dimension so there is no dimension mark to go along with it. Consequently, if we write 2 for the measure of an angle we understand that the angle is measured in radians. If we really mean an angle of 2 degrees, then we must write 2\(^\circ\).
Exercise \(\PageIndex{3}\)
Complete the following table to convert from degrees to radians and vice versa.
- Answer
-
Angle in Radians Angle in degrees \(0\) \(0^\circ\) \(\dfrac{\pi}{6}\) \(30^\circ\) \(\dfrac{\pi}{4}\) \(45^\circ\) \(\dfrac{\pi}{3}\) \(60^\circ\) \(\dfrac{\pi}{2}\) \(90^\circ\) \(\dfrac{7\pi}{6}\) \(210^\circ\) \(\dfrac{5\pi}{4}\) \(225^\circ\) \(\dfrac{4\pi}{3}\) \(240^\circ\) \(\dfrac{3\pi}{2}\) \(270^\circ\) \(\dfrac{5\pi}{3}\) \(300^\circ\) \(\dfrac{2\pi}{3}\) \(120^\circ\) \(\dfrac{4\pi}{3}\) \(135^\circ\) \(\dfrac{5\pi}{6}\) \(150^\circ\) \(\pi\) \(180^\circ\) \(\dfrac{7\pi}{4}\) \(315^\circ\) \(\dfrac{11\pi}{6}\) \(330^\circ\) \(2\pi\) \(360^\circ\)
Calculators and the Trigonometric Functions
We have now seen that when we think of \(\sin(t)\) or \(\cos(t)\), we can think of \((t)\) as a real number, the length of an arc, or the radian measure of an angle. We will see how to determine the exact values of the cosine and sine functions for a few special arcs (or angles). For example, we will see that \(\cos(\dfrac{\pi}{6}) = \dfrac{\sqrt{3}}{2}\). However, the definition of cosine and sine as coordinates of points on the unit circle makes it difficult to find exact values for these functions except at very special arcs . While exact values are always best, technology plays an important role in allowing us to approximate the values of the circular (or trigonometric)functions. Most hand-held calculators, calculators in phone or tablet apps, and online calculators have a cosine key and a sine key that you can use to approximate values of these functions, but we must keep in mind that the calculator only provides an approximation of the value, not the exact value (except for a small collection of arcs). In addition, most calculators will approximate the sine and cosine of angles. Remember to read homework questions carefully - if it says anything about an exact number, it probably does NOT want a calculator approximation.
| Angle in radians | Angle in degrees |
| 0 | \(0^\circ\) |
| \(\dfrac{\pi}{6}\) | |
| \(\dfrac{\pi}{4}\) | |
| \(\dfrac{\pi}{3}\) | |
| \(\dfrac{\pi}{2}\) | \(90^\circ\) |
| \(120^\circ\) | |
| \(\dfrac{3\pi}{4}\) | \(135^\circ\) |
| \(150^\circ\) | |
| \(180^\circ\) | |
| \(\dfrac{7\pi}{6}\) | |
| \(\dfrac{5\pi}{4}\) | |
| \(\dfrac{4\pi}{3}\) | |
| \(\dfrac{3\pi}{2}\) | \(270^\circ\) |
| \(300^\circ\) | |
| \(315^\circ\) | |
| \(330^\circ\) | |
| \(2\pi\) | \(360^\circ\) |
Table 1.1: Conversions between radians and degrees.
To do this, the calculator has two modes for angles: Radian and Degree. Because of the correspondence between real numbers, length of arcs, and radian measures of angles, for now, we will always put our calculators in radian mode. In fact, we have seen that an angle measured in radians subtends an arc of that radian measure along the unit circle. So the cosine or sine of an angle measure in radians is the same thing as the cosine or sine of a real number when that real number is interpreted as the length of an arc along the unit circle. (When we study the trigonometry of triangles, we will use the degree mode. In general, if a questions or application is given in degrees the solution is usually also given in degrees. Similarly, when working in radians the solution would also be given in radians, unless you have specifically been asked to convert from one to the other form.
Exercise \(\PageIndex{4}\)
We can use the Geogebra Applet called Terminal Points of Arcs on the Unit Circle at http://gvsu.edu/s/JY to approximate the values of the cosine and sine functions at certain values. For example, we could find the following:
- \(\cos(1) \approx 0.5403\), \(\sin(1) \approx 0.8415\).
- \(\cos(2) \approx -0.4161\), \(\sin(2) \approx 0.9093\).
- \(\cos(-4) \approx -0.6536\), \(\sin(-4) \approx 0.7568\).
- \(\cos(-15) \approx -0.7597\), \(\sin(-15) \approx -0.6503\).
Use a calculator to determine these values of the cosine and sine functions and compare the values to the ones above. Are they the same? How are they different?
- Answer
-
Using a calculator, we obtain the following results correct to ten decimal places:
\(\cos(1) \approx 0.5403023059, \sin(1) \approx 0.8414709848\). \(\cos(1) \approx 0:4161468365, \sin(1) \approx 0.9092974268\).The difference between these values and those obtained previously, is that these values are correct to 10 decimal places (and the others are correct to 4 decimal places). If we round off each of the values above to 4 decimal places, we get the same results.
- \(\cos(1) \approx 0:6536436209, \sin(1) \approx 0.7568024953\).
- \(\cos(1) \approx 0:7596879129, \sin(1) \approx 0.6502878402\).
Exercise \(\PageIndex{5}\)
When working with sine and cosine we are reminded that these two functions can be used to generate the remaining 4 functions - tangent, secant, cosecant and cotangent.
Each of these trigonometric functions are positive in different quadrants of the unit circle. Now that we can calculate the values of sine and cosine at ANY terminal point along a unit circle, we now need to be able to do the same for the remaining 4 functions.
- In quadrant I A LL trigonometric functions are positive.
- In quadrant II the S INE function is positive.
- In quadrant III the T ANGENT function is positive.
- In quadrant IV the C OSINE function is positive.
Using the first letter from each of these quadrants and putting it into an easy-to-remember phrase can help you remember where functions are positive.
Here are some common phrases: " A ll S tudents T ake C alculus" " A S mart T rig C lass" or maybe you can think up a better one!
- In what quadrant does the terminal ray reside if \(sin(t)<0\) and \(cos(t)<0\)?
- In what quadrant does the terminal ray reside if \(tan(t)<0\) and \(cos(t)>0\)?
- In what quadrant does the terminal ray reside if \(sec(t)<0\) and \(tan(t)>0\)?
- In what quadrant does the terminal ray reside if \(csc(t)>0\) and \(cot(t)<0\)?
- Answer
-
- Sine is negative in quadrant III and IV, while cosine is negative in quadrant II and III. Therefore the overlap must be in quadrant II where both functions are negative.
- Recall that tangent has an identity where \(tan(t)=\dfrac{sin(t)}{cos(t)}\) is negative in quadrant II and IV, while cosine is positive in quadrant I and IV. Therefore the overlap must be in quadrant IV where these parameters are both met.
- Recall that secant has an identity where \(sec(t)=\dfrac{1}{cos(t)}\) this implies that the same quadrants will be positive for secant as for cosine. Secant is negative in quadrants II and III, while tangent is positive in quadrant I and III. Therefore the overlap must be in quadrant III where these parameters are both met.
- Recall that cosecant has an identity where \(csc(t)=\dfrac{1}{sin(t)}\) this implies that the same quadrants will be positive for cosecant as for sine. Recall that cotangent is the reciprocal of tangent and has an identity where \(cot(t)=\dfrac{cos(t)}{sin(t)}\) is negative in quadrant II and IV. Cosecant is positive in quadrants I and II, while cotangent is negative in quadrant II and IV. Therefore, the overlap must be in quadrant II where these parameters are both met.
Exercise \(\PageIndex{6}\)
Sometimes it will be important to write a given trigonometric function in terms of another.
The Pythagorean Theorem is \(sin^2(\theta)\)+\(cos^2(\theta)\)=1.
Dividing this initial identity by \(sin^2(\theta)\) will generate another form for Pythagorean Theorem.
Dividing this initial identity by \(cos^2(\theta)\) will generate a third form.
Use these three different forms to find the functions listed in terms of another function. Use the quadrant to determine if the function is positive or negative.
Remember from the previous exercise that the following are true:
- In quadrant I A LL trigonometric functions are positive.
- In quadrant II the S INE function is positive.
- In quadrant III the T ANGENT function is positive.
- In quadrant IV the C OSINE function is positive.
Use the Pythagorean Theorem and the phrase you have memorized to find the following trigonometric functions:
- Write the \(tan\theta\) in terms of \(cos\theta\) for \(\theta\) in Quadrant II.
- Write the \(sin\theta\) in terms of \(cos\theta\) for \(\theta\) in Quadrant III.
- Write the \(tan\theta\) in terms of \(sec\theta\) for \(\theta\) in Quadrant I.
- Write the \(sec\theta\) in terms of \(cos\theta\) for \(\theta\) in Quadrant IV.
- Answer
-
- We need to find a way to write \(tan\theta\) equal to something on the right with only cosine functions included. If we divide \(sin^2\theta\)+\(cos^2\theta\)=1 by \(cos^2\theta\) we will have a form for the Pythagorean Theorem in terms of tangent and cosine. \[\dfrac{sin^2\theta}{cos^2\theta}+\dfrac{cos^2\theta}{cos^2\theta}=\dfrac{1}{cos^2\theta}\] Now, solving for tangent we should have: \[tan^2\theta=\dfrac{1}{cos^2\theta}-\dfrac{cos^2\theta}{cos^2\theta}\] Simplify and take the square root of both sides. Remember, tangent is negative in Quadrant II, therefore our answer should be: \[tan\theta=-\dfrac{\sqrt{1-cos^2 \theta} }{cos\theta }\]
- \[sin\theta=- \sqrt{1-cos^2\theta} \]
- \[tan\theta= \sqrt{sec^2\theta-1} \]
- \[tan\theta=-\dfrac{1}{ \sqrt{1-cos^2 \theta } }\]
Exercise \(\PageIndex{7}\)
Throughout trigonometry - even in Calculus, Linear Algebra and some Differential Equations classes - we will be asked to find the terminal point given limited information and sometimes restrictions on your domain or quadrant. Use the Pythagorean Theorem to find the missing sides and recognize when the function will be positive or negative based on the quadrant. It is usually helpful to first restrict the quadrant to the one in which the values are true, then calculate the exact value for the remaining five functions based on the given information.
- Given \(sin\theta=-\dfrac{4}{5}\) find the remaining five values for the remaining five functions if \(\theta\) is in Quadrant IV.
- Given \(cos\theta=-\dfrac{5}{6}\) find the remaining five values for the remaining five functions if \(cot\theta>0\).
- Given \(csc\theta=3\) find the remaining five values for the remaining five functions in Quadrant I.
- Given \(cos\theta=\dfrac{7}{13}\) find the remaining five values for the remaining five functions if \(sin\theta<0\).
- Given \(sec\theta=-\dfrac{1}{2}\) find the remaining five values for the remaining five functions if \(csc\theta<0\).
- Given \(cos\theta=-\dfrac{2}{11}\) find the remaining five values for the remaining five functions if\(tan\theta<0\).
- Answer
-
- Rotating through to a terminal ray in Quadrant IV, we now can draw a triangle using the terminal ray as the hypotenuse of the triangle. Use a reference angle off the x axis to orient the opposite and adjacent sides. The missing side is the x-direction length. Using the Pythagorean Theorem the sides of the triangle should be: Opposite is -4, Adjacent is 3 and Hypotenuse is 5. Now, using the identities, \[sin\theta=\dfrac{opposite}{hypotenuse}=\dfrac{-4}{5}\] \[cos\theta=\dfrac{adjacent}{hypotenuse}=\dfrac{3}{5}\] and \[tan\theta=\dfrac{opposite}{adjacent}=\dfrac{sin\theta}{cos\theta}=\dfrac{-4}{3}\] For the remaining three trig functions we can use what we found previously, just as reciprocals: \[csc\theta=\dfrac{hypotenuse}{opposite}=\dfrac{5}{-4}\] \[sec\theta=\dfrac{hypotenuse}{adjacent}=\dfrac{5}{3}\] \[cot\theta=\dfrac{adjacent}{opposite}=\dfrac{cos\theta}{sin\theta}=\dfrac{3}{-4}\] NOTE:From here on the answers will be simplified such that we bring the negative sign in front of the fraction and it is understood that it was associated with the negative side of the triangle. It is more formal to write it in front of the fraction instead of with the numerator or denominator.
- Cotangent is only positive in Quadrants I and III. While Cosine is only negative in Quadrants II and III. Therefore, we know our values for all six trig functions must exist in Quadrant III! \[sin\theta=\dfrac{opposite}{hypotenuse}=-\dfrac{\sqrt{11}}{6}\] \[cos\theta=-\dfrac{5}{6}\] \[tan\theta=\dfrac{opposite}{adjacent}=\dfrac{sin\theta}{cos\theta}=\dfrac{\sqrt{11}}{5}\] For the remaining three trig functions we can use what we found previously, just as reciprocals:\[csc\theta=\dfrac{hypotenuse}{opposite}=-\dfrac{6\sqrt{11}}{11}\] \[sec\theta=\dfrac{hypotenuse}{adjacent}=-\dfrac{6}{5}\] \[cot\theta=\dfrac{adjacent}{opposite}=\dfrac{cos\theta}{sin\theta}=\dfrac{5\sqrt{11}}{11}\]
- \[sin\theta=\dfrac{1}{3}\] \[cos\theta=\dfrac{2\sqrt{2}}{3}\] \[tan\theta=\dfrac{\sqrt{2}}{4}\] \[csc\theta=3\] \[sec\theta=\dfrac{3\sqrt{2}}{4}\] \[cot\theta=2\sqrt{2}\]
- \[sin\theta=-\dfrac{2\sqrt{30}}{13}\] \[cos\theta=\dfrac{7}{13}\] \[tan\theta=2\dfrac{2\sqrt{30}}{7}\] \[csc\theta=-\dfrac{13\sqrt{30}}{60}\] \[sec\theta=\dfrac{13}{7}\] \[cot\theta=-\dfrac{7\sqrt{30}}{60}\]
- Algebraically, we can calculate the value of cosine because \(sec\theta=-\dfrac{1}{2}\). Therefore, cosine is -2. If we try to calculate the values for the remaining 5 functions we notice that the hypotenuse is less than the adjacent side. This should be a warning sign that the values do not exist. Geometrically/Graphically, we could plot or graph the function \(f(x)=sec\theta\) simultaneously with \(f(x)=-\dfrac{1}{2}\) to see that there is never a point at which the two functions intersect. This should lead us to believe that there is NO SOLUTION, or a solution DOES NOT EXIST (sometimes annotated as DNE).
- \[sin\theta=\dfrac{3\sqrt{13}}{11}\] \[cos\theta=-\dfrac{2}{1}\] \[tan\theta=-\dfrac{3\sqrt{13}}{2}\] \[csc\theta=\dfrac{11\sqrt{13}}{39}\] \[sec\theta=-\dfrac{11}{2}\] \[cot\theta=-\dfrac{2\sqrt{13}}{39}\]
Section Summary
In this section, we studied the following important concepts and ideas:
- An angle is formed by rotating a ray about its endpoint. The ray in its initial position is called the initial side of the angle, and the position of the ray after it has been rotated is called the terminal side of the ray. The endpoint of the ray is called the vertex of the angle .
- When the vertex of an angle is at the origin in the \(xy\)-plane and the initial side lies along the positive x-axis, we see that the angle is in standard position .
- There are two ways to measure angles. For degree measure, one complete wrap around a circle is 360 degrees, denoted 360\(^\circ\). The unit measure of 1\(^\circ\) is an angle that is 1=360 of the central angle of a circle. An angle of one radian is the angle in standard position on the unit circle that is subtended by an arc of length 1 (in the positive direction).
- We convert the measure of an angle from degrees to radians by using the fact that each degree is radians. We convert the measure of an angle from 180 radians to degrees by using the fact that each radian is 180 degrees.
- We looked at the strategies to use when finding the range values for all 6 trigonometric functions.
- We imposed restriction on the domain and quadrant, then found the missing trig function value. This will come in handy in the following sections!


3.1: Use a Problem-Solving Strategy
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Learning Objectives
By the end of this section, you will be able to:
- Approach word problems with a positive attitude
- Use a problem-solving strategy for word problems
- Solve number problems
Note
Before you get started, take this readiness quiz.
-
Translate “6 less than twice
x
” into an algebraic expression.
If you missed this problem, review Exercise 1.3.43 . -
Solve: \(\frac{2}{3}x=24\).
If you missed this problem, review Exercise 2.2.10 . -
Solve: \(3x+8=14\).
If you missed this problem, review Exercise 2.3.1 .
 Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion? How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below (Figure \(\PageIndex{1}\))?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure \(\PageIndex{2}\) and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
USE A PROBLEM-SOLVING STRATEGY TO SOLVE WORD PROBLEMS.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
Example \(\PageIndex{1}\)
Pilar bought a purse on sale for \($18\), which is one-half of the original price. What was the original price of the purse?
Solution
Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.
In this problem, is it clear what is being discussed? Is every word familiar?Let p = the original price of the purse.
Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
In this problem, the words “what was the original price of the purse” tell us what we need to find.
Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.
Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.
| Restate the problem in one sentence with all the important information. | \(\color{cyan} \underbrace{\strut \color{black}\mathbf{18}} \quad \underbrace{\strut \color{black}\textbf{ is }} \quad \underbrace{\color{black}\textbf{one-half the original price.}}\) |
| Translate into an equation. | \(18 \qquad = \qquad \qquad \qquad \frac{1}{2}\cdot p\) |
Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.
| Solve the equation. | \(18 = \frac{1}{2}p\) |
| Multiply both sides by 2. | \( {\color{red}{2}}\cdot 18 = {\color{red}{2}}\cdot \frac{1}{2}p \) |
| Simplify. | \(36 = p\) |
Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that \(p=36\),which means “the original price” was \($36\).
Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.If this were a homework exercise, our work might look like this:
Pilar bought a purse on sale for \($18\), which is one-half the original price. What was the original price of the purse?
Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”
The answer to the question is: “The original price of the purse was $36.”| Let \(p =\) the original price. | |
| \(18\) is one-half the original price. | |
| \(18 = \frac{1}{2}p\) | |
| Multiply both sides by \(2\). | \( {\color{red}{2}}\cdot 18 = {\color{red}{2}}\cdot \frac{1}{2}p \) |
| Simplify. | \(36 = p\) |
| Check. Is \($36\) a reasonable price for a purse? | |
| Yes. | |
| Is \(18\) one half of \(36\)? | |
| \(18 \stackrel{?}{=} \frac{1}{2}\cdot 36\) | |
| \(18 = 18\checkmark\) | |
| The original price of the purse was \($36\). |
Try It \(\PageIndex{2}\)
Joaquin bought a bookcase on sale for \($120\), which was two-thirds of the original price. What was the original price of the bookcase?
- Answer
-
\($180\)
Try It \(\PageIndex{3}\)
Two-fifths of the songs in Mariel’s playlist are country. If there are \(16\) country songs, what is the total number of songs in the playlist?
-
Answer -
\(40\)
Let’s try this approach with another example.
Example \(\PageIndex{4}\)
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were \(11\) girls in the study group. How many boys were in the study group?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | How many boys were in the study group? |
| Step 3. Name. Choose a variable to represent the number of boys. | Let \(n=\) the number of boys. |
| Step 4. Translate. Restate the problem in one sentence with all the important information. | \(\color{cyan} \underbrace{\color{black}\textbf{The number}\\ \color{black}\textbf{of girls}(11)} \quad \underbrace{\strut \text{ } \\ \color{black}\textbf{was}} \quad \underbrace{\color{black}\textbf{three more than}\\ \color{black}\textbf{twice the number of boys}}\) |
| Translate into an equation. | \(\qquad 11 \qquad \quad = \qquad \qquad \quad 2b + 3\) |
| Step 5. Solve the equation. | \(\quad 11 = 2b + 3 \) |
| Subtract 3 from each side. | \(\quad 11 \,{\color{red}{- \,3}} = 2b + 3 \,{\color{red}{- \,3}} \) |
| Simplify. | \(\quad 8 = 2b \) |
| Divide each side by 2. | \( \quad \dfrac{8}{\color{red}{2}}=\dfrac{2b}{\color{red}{2}} \) |
| Simplify. | \(\quad 4 = b\) |
| Step 6. Check. First, is our answer reasonable? Yes, having \(4\) boys in a study group seems OK. The problem says the number of girls was \(3\) more than twice the number of boys. If there are four boys, does that make eleven girls? Twice \(4\) boys is \(8\). Three more than \(8\) is \(11\). | |
| Step 7. Answer the question. | There were \(4\) boys in the study group. |
Try It \(\PageIndex{5}\)
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was \(3\) more than twice the number of notebooks. He bought \(7\) textbooks. How many notebooks did he buy?
- Answer
-
\(2\)
Try It \(\PageIndex{6}\)
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed \(22\) Sudoku puzzles. How many crossword puzzles did he do?
- Answer
-
\(7\)
Solve Number Problems
Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
Example \(\PageIndex{7}\)
The difference of a number and six is \(13\). Find the number.
Solution
| Step 1. Read the problem. Are all the words familiar? | |
| Step 2. Identify what we are looking for. | the number |
| Step 3. Name. Choose a variable to represent the number. | Let \(n=\) the number. |
| Step 4. Translate. Remember to look for clue words like "difference... of... and..." | |
| Restate the problem as one sentence. | \(\color{cyan} \underbrace{\color{black}\textbf{The difference of the number and }\mathbf{6}} \quad \underbrace{\strut \color{black}\textbf{ is }} \quad \underbrace{\strut \color{black}\mathbf{13}}\) |
| Translate into an equation. | \(\qquad \qquad \qquad n-6 \qquad \qquad \qquad \quad = \quad 13\) |
| Step 5. Solve the equation. | \(\quad n - 6 = 13\) |
| Simplify. | \(\quad n =19\) |
| Step 6. Check. | |
| The difference of \(19\) and \(6\) is \(13\). It checks! | |
| Step 7. Answer the question. | The number is \(19\). |
Try It \(\PageIndex{8}\)
The difference of a number and eight is \(17\). Find the number.
- Answer
-
\(25\)
Try It \(\PageIndex{9}\)
The difference of a number and eleven is \(−7\). Find the number.
- Answer
-
\(4\)
Example \(\PageIndex{10}\)
The sum of twice a number and seven is \(15\). Find the number.
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the number |
| Step 3. Name. Choose a variable to represent the number. | Let \(n =\) the number. |
| Step 4. Translate. | |
| Restate the problem as one sentence. | |
| Translate into an equation. | |
| Step 5. Solve the equation. | |
| Subtract 7 from each side and simplify. | |
| Divide each side by 2 and simplify. | |
| Step 6. Check. | |
| Is the sum of twice 4 and 7 equal to 15? | |
| \(\begin{array} {rrl} {2\cdot 4 + 7} &{\stackrel{?}{=}}& {15} \\ {15} &{=} &{15\checkmark} \end{array}\) | |
| Step 7. Answer the question. | The number is \(4\). |
Try It \(\PageIndex{11}\)
The sum of four times a number and two is \(14\). Find the number.
- Answer
-
\(3\)
Try It \(\PageIndex{12}\)
The sum of three times a number and seven is \(25\). Find the number.
- Answer
-
\(6\)
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Example \(\PageIndex{13}\)
One number is five more than another. The sum of the numbers is 21. Find the numbers.
Solution
| Step 1. Read the problem. | ||
| Step 2. Identify what we are looking for. | We are looking for two numbers. | |
| Step 3. Name. We have two numbers to name and need a name for each. | ||
| Choose a variable to represent the first number. | Let \(n=1^{st}\) number. | |
| What do we know about the second number? | One number is five more than another. | |
| \(n+5=2^{nd}\) number | ||
| Step 4. Translate. Restate the problem as one sentence with all the important information. | The sum of the 1 st number and the 2 nd number is 21. | |
| Translate into an equation. | ||
| Substitute the variable expressions. | ||
| Step 5. Solve the equation. | ||
| Combine like terms. | ||
| Subtract 5 from both sides and simplify. | ||
| Divide by 2 and simplify. | ||
| Find the second number, too. | ||
| Step 6. Check. | ||
| Do these numbers check in the problem? | ||
| Is one number \(5\) more than the other? | \(13\stackrel{?}{=} 8 + 5\) | |
| Is thirteen \(5\) more than \(8\)? Yes. | \(13 = 13\checkmark\) | |
| Is the sum of the two numbers \(21\)? | \(8 + 13 \stackrel{?}{=} 21\) | |
| \(21 = 21\checkmark\) | ||
| Step 7. Answer the question. | The numbers are \(8\) and \(13\). |
Try It \(\PageIndex{14}\)
One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.
- Answer
-
9, 15
Try It \(\PageIndex{15}\)
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.
- Answer
-
27, 31
Example \(\PageIndex{16}\)
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
Solution
| Step 1. Read the problem. | ||
| Step 2. Identify what we are looking for. | We are looking for two numbers. | |
| Step 3. Name. | ||
| Choose a variable. | Let \(n=1^{st}\) number. | |
| One number is 4 less than the other. | \(n−4=2^{nd}\) number | |
| Step 4. Translate. | ||
| Write as one sentence. | The sum of the 2 numbers is negative 14. | |
| Translate into an equation. | ||
| Step 5. Solve the equation. | ||
| Combine like terms. | ||
| Add 4 to each side and simplify. | ||
| Simplify. | ||
| Step 6. Check. | ||
| Is −9 four less than −5? | \(-5-4\stackrel{?}{=}-9\) | |
| \(-9 = -9 \checkmark\) | ||
| Is their sum −14? | \(-5+ (-9)\stackrel{?}{=}-14\) | |
| \(-14 = -14 \checkmark\) | ||
| Step 7. Answer the question. | The numbers are −5 and −9. |
Try It \(\PageIndex{17}\)
The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.
- Answer
-
-15, -8
Try It \(\PageIndex{18}\)
The sum of two numbers is \(−18\). One number is \(40\) more than the other. Find the numbers.
- Answer
-
-29, 11
Example \(\PageIndex{19}\)
One number is ten more than twice another. Their sum is one. Find the numbers.
Solution
| Step 1. Read the problem. | ||
| Step 2. Identify what you are looking for. | We are looking for two numbers. | |
| Step 3. Name. | ||
| Choose a variable. | Let \(x=1^{st}\) number. | |
| One number is 10 more than twice another. | \(2x+10=2^{nd}\) number | |
| Step 4. Translate. | ||
| Restate as one sentence. | Their sum is one. | |
| The sum of the two numbers is 1. | ||
| Translate into an equation. | ||
| Step 5. Solve the equation. | ||
| Combine like terms. | ||
| Subtract 10 from each side. | ||
| Divide each side by 3. | ||
| Step 6. Check. | ||
| Is ten more than twice −3 equal to 4? | \(2(-3) + 10 \stackrel{?}{=} 4\) | |
| \(-6 + 10 \stackrel{?}{=} 4\) | ||
| \(4 = 4\checkmark\) | ||
| Is their sum 1? | \(-3 + 4 \stackrel{?}{=} 1\) | |
| \(1 = 1\checkmark\) | ||
| Step 7. Answer the question. | The numbers are −3 and 4. |
Try It \(\PageIndex{20}\)
One number is eight more than twice another. Their sum is negative four. Find the numbers.
- Answer
-
\(-4,\; 0\)
Try It \(\PageIndex{21}\)
One number is three more than three times another. Their sum is \(−5\). Find the numbers.
- Answer
-
\(-3,\; -2\)
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
\[\begin{array}{l}{1,2,3,4} \\ {-10,-9,-8,-7} \\ {150,151,152,153}\end{array}\]
Notice that each number is one more than the number preceding it. So if we define the first integer as \(n\), the next consecutive integer is \(n+1\). The one after that is one more than \(n+1\), so it is \(n+1+1\), which is \(n+2\).
\[\begin{array}{ll}{n} & {1^{\text { st }} \text { integer }} \\ {n+1} & {2^{\text { nd }} \text { consecutive integer }} \\ {n+2} & {3^{\text { rd }} \text { consecutive integer } \ldots \text { etc. }}\end{array}\]
Example \(\PageIndex{22}\)
The sum of two consecutive integers is \(47\). Find the numbers.
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what you are looking for. | two consecutive integers |
| Step 3. Name each number. | Let \(n=1^{st}\) integer. |
| \(n+1=\) next consecutive integer | |
| Step 4. Translate. | |
| Restate as one sentence. | The sum of the integers is \(47\). |
| Translate into an equation. | |
| Step 5. Solve the equation. | |
| Combine like terms. | |
| Subtract 1 from each side. | |
| Divide each side by 2. | |
| Step 6. Check. | |
| \(\begin{array} {lll} {23 + 24} &{\stackrel{?}{=}} &{47} \\ {47} &{=} &{47\checkmark} \end{array}\) | |
| Step 7. Answer the question. | The two consecutive integers are 23 and 24. |
Try It \(\PageIndex{23}\)
The sum of two consecutive integers is 95. Find the numbers.
- Answer
-
47, 48
Try It \(\PageIndex{24}\)
The sum of two consecutive integers is −31. Find the numbers.
- Answer
-
-16, -15
Example \(\PageIndex{25}\)
Find three consecutive integers whose sum is −42.
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | three consecutive integers |
| Step 3. Name each of the three numbers. | Let \(n=1^{st}\) integer. |
| \(n+1= 2^{nd}\) consecutive integer | |
| \(n+2= 3^{rd}\) consecutive integer | |
| Step 4. Translate. | |
| Restate as one sentence. | The sum of the three integers is \(−42\). |
| Translate into an equation. | |
| Step 5. Solve the equation. | |
| Combine like terms. | |
| Subtract 3 from each side. | |
| Divide each side by 3. | |
|
|
|
|
|
|
| Step 6. Check. | |
| \(\begin{array}{lll} {-13 + (-14) + (-15)} &{\stackrel{?}{=}} &{-42} \\ {-42} &{=} &{-42\checkmark} \end{array}\) | |
| Step 7. Answer the question. | The three consecutive integers are −13, −14, and −15. |
Try It \(\PageIndex{26}\)
Find three consecutive integers whose sum is −96.
- Answer
-
-33, -32, -31
Try It \(\PageIndex{27}\)
Find three consecutive integers whose sum is −36.
- Answer
-
-13, -12, -11
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
\[\begin{array}{l}{18,20,22} \\ {64,66,68} \\ {-12,-10,-8}\end{array}\]
Notice each integer is \(2\) more than the number preceding it. If we call the first one \(n\), then the next one is \(n+2\). The next one would be \(n+2+2\) or \(n+4\).
\[\begin{array}{cll}{n} & {1^{\text { st }} \text { even integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive even integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive even integer } \ldots \text { etc. }}\end{array}\]
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers \(77\), \(79\), and \(81\).
\[\begin{array}{l}{77,79,81} \\ {n, n+2, n+4}\end{array}\]
\[\begin{array}{cll}{n} & {1^{\text { st }} \text {odd integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive odd integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive odd integer } \ldots \text { etc. }}\end{array}\]
Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.
Example \(\PageIndex{28}\)
Find three consecutive even integers whose sum is 84.
Solution
\[\begin{array}{ll} {\textbf{Step 1. Read} \text{ the problem.}} & {} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} & {\text{three consecutive even integers}} \\ {\textbf{Step 3. Name} \text{ the integers.}} & {\text{Let } n = 1^{st} \text{ even integers.}} \\ {} &{n + 2 = 2^{nd} \text{ consecutive even integer}} \\ {} &{n + 4 = 3^{rd} \text{ consecutive even integer}} \\ {\textbf{Step 4. Translate.}} &{} \\ {\text{ Restate as one sentence. }} &{\text{The sum of the three even integers is 84.}} \\ {\text{Translate into an equation.}} &{n + n + 2 + n + 4 = 84} \\ {\textbf{Step 5. Solve} \text{ the equation. }} &{} \\ {\text{Combine like terms.}} &{n + n + 2 + n + 4 = 84} \\ {\text{Subtract 6 from each side.}} &{3n + 6 = 84} \\ {\text{Divide each side by 3.}} &{3n = 78} \\ {} &{n = 26 \space 1^{st} \text{ integer}} \\\\ {} &{n + 2\space 2^{nd} \text{ integer}} \\ {} &{26 + 2} \\ {} &{28} \\\\ {} &{n + 4\space 3^{rd} \text{ integer}} \\ {} &{26 + 4} \\ {} &{30} \\ {\textbf{Step 6. Check.}} &{} \\\\ {26 + 28 + 30 \stackrel{?}{=} 84} &{} \\ {84 = 84 \checkmark} & {} \\ {\textbf{Step 7. Answer} \text{ the question.}} &{\text{The three consecutive integers are 26, 28, and 30.}} \end{array}\]
Try It \(\PageIndex{29}\)
Find three consecutive even integers whose sum is 102.
- Answer
-
32, 34, 36
Try It \(\PageIndex{30}\)
Find three consecutive even integers whose sum is −24.
- Answer
-
−10,−8,−6
Example \(\PageIndex{31}\)
A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | How much does the husband earn? |
| Step 3. Name . | |
|
Choose a variable to represent the amount
the husband earns. |
Let \(h=\) the amount the husband earns. |
| The wife earns \($16,000\) less than twice that. | \(2h−16,000\) the amount the wife earns. |
| Step 4. Translate. | Together the husband and wife earn \($110,000\). |
|
Restate the problem in one sentence with
all the important information. |
|
| Translate into an equation. | |
| Step 5. Solve the equation. | \(h + 2h − 16,000 = 110,000\) |
| Combine like terms. | \(3h − 16,000 = 110,000\) |
| Add \(16,000\) to both sides and simplify. | \(3h = 126,000\) |
| Divide each side by \(3\). | \(h = 42,000\) |
| \($42,000\) amount husband earns | |
| \(2h − 16,000\) amount wife earns | |
| \(2(42,000) − 16,000\) | |
| \(84,000 − 16,000\) | |
| \(68,000\) | |
| Step 6. Check. | |
| If the wife earns \($68,000 \) and the husband earns \($42,000 \) is the total \($110,000 \) ? Yes! | |
| Step 7. Answer the question. | The husband earns \($42,000 \) a year. |
Try It \(\PageIndex{32}\)
According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?
- Answer
-
$5000
Try It \(\PageIndex{33}\)
U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?
- Answer
-
$19300
Key Concepts
-
Problem-Solving Strategy
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
-
Consecutive Integers
Consecutive integers are integers that immediately follow each other.\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+1} & {2^{\text { nd }} \text {consecutive integer }} \\ {n+2} & {3^{\text { rd }} \text { consecutive integer } \ldots \text { etc. }}\end{array}\]
Consecutive even integers are even integers that immediately follow one another.\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive even integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive even integer } \ldots \text { etc. }}\end{array}\]
Consecutive odd integers are odd integers that immediately follow one another.\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive odd integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive odd integer } \ldots \text { etc. }}\end{array}\]


19.1: Introduction - The Turkey Boat Problem
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- Last updated
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Each spring an excited group of athletic individuals filter into the small and close-knit whitewater rafting community. Upon arriving at the river community these trainees are given a place to camp, and an opportunity to ride along with the senior guiding staff. Although the senior guides are tolerant of the new arrivals, they recognize that, in an average training year, only one third of them will eventually become guides.
Over several months, the new arrivals will be asked to practise rescue techniques, learn to read complex whitewater hydraulics and develop sound decision-making abilities within highly stressful and quickly changing conditions.
Only after these hard skills have been attained can trainees begin to comprehend the immense responsibility they hold for the safety of others. In an effort to manage the risks inherent in rafting, guides need to develop their problem-solving techniques before they progress into dangerous situations. Whitewater guides must understand that each participant plays an important role in an interdependent team, which must coordinate efforts in order to successfully navigate treacherous and complex whitewater rapids.
Often people who want to become a professional guide see the role of guiding as a burden to be shouldered through physical strength and expertise. Indeed, physical competence is an important component however, the importance of the other team members within the boat is often underestimated. This lack of recognition can quickly lead to failures in communication— creating an environment where accidents can occur.
The turkey boat is used in raft guide training to address the importance of teamwork and communication. A team of guide hopefuls (around eight, total) with little or no direct advice, are given a whitewater raft, paddles, life jackets and helmets and told to navigate difficult whitewater rapids.
Much like learning to drive a car, the kinetics of whitewater rafting are not overly complex. Your paddle works as the steering wheel, gas pedal and brake. And as with driving a car, once the basic kinetics have been learned, confidence quickly follows. Unlike a car, however, a raft can be controlled by any of the individuals holding a paddle—and to complicate matters, each individual is viewing the river from a different location. So the turkey boat consists of eight individuals, armed with minimal technical knowledge and growing (sometimes inflated) confidence, eagerly striving to prove their leadership ability.
Leadership development schools such as Outward Bound have taught basic sports psychology for years. The idea behind these schools is that group development progresses through stages. Further, these stages can be used to develop tools that facilitate highly dynamic group interactions. Although raft guiding activities can be classified as adventure recreation, or perhaps even educational, the focus of guide training programs are to build physical and social competencies, including guiding technique, communication, respect, and problem-solving. Because of this focus, guide training programs can be described as a developmental adventure education.
Developmental adventure education contains a strong process-based component. This process can be used to assess the goals of the group and attempt to facilitate a trajectory of learning for the participants. These trajectories work to place engagement in activities in the context of a valued future within the group. In this way, learning can be experienced as a form of identity.
The adaptation of personal behaviour is the foundation of developmental adventure education. Facilitating the process of adaptation guides the participant in developing an understanding of group expectations, norms and behaviours.


15.1: Plate Tectonics and Continental Drift
Introduction
The surface of the lithosphere is fractured into a number of tectonic plates (also known as lithospheric or crustal plates) which are in constant motion. As these plates move and collide, the lithosphere buckles, warps, and is torn apart. When this occurs, the Earth's surface shakes with great force, like that which accompanies earthquakes. Volcanoes are common along many plate boundaries as well.
Plate Tectonics
Plate tectonics refers to the process of plate formation, movement, and destruction. It finds its foundations in two theories, continental drift and sea-floor spreading. Continental drift describes the movements of continents over the Earth's surface. Sea-floor spreading refers to the creation new oceanic plate material and movement away from the mid-ocean ridge. It was Alfred Wegener in the early 1900's who brought forth the concept that the "shell" of the Earth's surface was fractured, and these "pieces" drifted about. Blasphemy in the minds of scientists of Wegener's day, some 50 years later his ideas were finally accepted. Wegener was able to piece together (pardon the pun) several bits of information which led to his conclusion that the present configuration of the continents is not the same as it was in the past. In fact, the continents were one "super-continent" called Pangea .
Look at a map of the Earth like that in Figure \(\PageIndex{3}\). Carefully examine the east coast of South America and then let your eyes drift to the west coast of Africa. It looks like you could "fit" South America up against Africa like a puzzle. The same can be said for the fit between North America, Africa, and Europe (Figure \(\PageIndex{4}\))
When we slide the continents together, some over lap between the land masses occurs. This is possibly due to the creation of exotic terrain , new land that has been formed somewhere else and moved to its present location. This remarkable correspondence provides circumstantial evidence for the theory of continental drift.
Looking for the evidence
If the continents were in one piece at some time in the past, we should find similar fossils and rocks on both continents which is precisely what Wegener discovered. By studying the geologic record, the fossil record, and climatic record, he found remarkable similarities between Africa and South America.
Fossils of the same species of plants and animals were found in similar geologic formations in different parts of the world, most notably South America, Africa, and India. For example, fossils of the Glossopteris, an ancient fern, are found in South America, Africa, Antarctica, India, and Australia. It was hypothesized that such a distribution could only come about if the continents were all part of the one super-continent.
Examining the stratigraphy (vertical sequences) of the rock record, Wegener could point to further evidence for Pangea and continental drift. Wegener noted that the rock sequences in South America, Africa, India, and Australia are very similar. Wegener showed that the same three bottoms layers occurred on each of the continents. The bottom layer, called tillite, was thought to be of glacial origin. The middle layer composed of coal beds, shale and sandstone contained Glossopteris fossils, as did the bottom tillite layer. The top most and youngest layer is lava flows. Such a strong similarity in the rock record of these localities, now separated by great geographic distance, lent credence to Wegener's notion of continental drift.
The occurrence of glacial features (Figure \(\PageIndex{7}\)) in the geologic record of South America, Africa, India ,and Australia provides further evidence for the notion of continental drift. Glaciers affected all or part of these continents at the same time in Earth history.
Video : Pangea (Courtesy Khan Academy)
The Mobile Crust
Wegener's ideas were not readily accepted during his day because he did not offer a plausible mechanism for the movement of the continents. Wegener suggested it was the spin of the Earth that caused plates to "plow" their way through the mantle beneath (Figure \(\PageIndex{9}\)).
Interestingly, there appeared to be a split in the scientific community in a geographical sense at the time. Geologists in the Northern hemisphere were less accepting as those living in the Southern hemisphere who were familiar with the rocks Wegener used to support his hypothesis. It wasn't until 1928 that a Scottish geologist proposed a mechanism to drive continental drift. Arthur Holmes believed a fluid mantle possessed convection currents created by heat trapped beneath the Earth's surface. Holmes hypothesized that convection currents welled up toward the surface and then drug continents across the surface.
In the late 1950's and early 1960's oceanographic research was opening the final frontier on Earth, the mysteries of the ocean floor. During the 1950's seismologists showed that earthquake activity was concentrated along the longest continuous mountain system on Earth, the mid-ocean ridge. Known for over a century, the mid-ocean ridge system in the Atlantic Ocean rises some 6,500 feet above the surrounding ocean floor and extends for more that 37,500 miles (60,000 km) in all the world's oceans. At the crest of the ridge system lies a trough or rift. These rifts are about 20 miles wide (30 km) and 6,500 feet (2,900 m) deep and are a site where lava is expelled on to the ocean floor. The youngest material is found near the ridge with rocks of increasing age further away. It appears that the mid-ocean ridge is the site of sea floor spreading leading to plate movement. The key to sea floor spreading was found in the magnetic properties of rock lying on the sea floor.
Geoscientists found that as new material is extruded from the mid-ocean ridge onto the ocean floor, the polarity of the rocks is frozen in time. Dragging a magnetometer across the ridge, they noticed that the polarity recorded a series of "flips" when the polarity of the rocks reversed. Parallel stripes of magnetic reversals on either side of the ridge was supporting evidence for sea floor spreading. ( See animation of magnetic stripe formation - caution large file) ( For more details see: North Cascades Geology - Sea Floor spreading" and Magnetic Stripes and isotope clocks (USGS). These ideas set the stage for further research that has given us a plausible explanation for the movement of the crust and the drifting of continents first proposed by Wegener.
Modern Theory of Plate Movement and Continental Drift
The movement of lithospheric plates referred to as continental drift , is believed to be caused by the radioactive decay of elements in the core and mantle that produces heat. The heat in turn creates convection currents in the mantle which "drive" the plates along their path of movement. When plates collide, heavier, more dense plates dive beneath lighter, less dense crustal plates along subduction zones . As the heavier plate moves downward into the mantle, the increase in temperature and pressure drive water and other volatile fluids ("dewatering") from the oceanic crust. At a depth of 100 kilometers (60 miles), the water-rich fluids decrease the melting point of mantle rock causing it to melt. The magma that slowly moves upwards and may be extruded onto the surface as lava (Figure \(\PageIndex{13}\)). Some plates slip past one another, creating earthquakes, (Figure \(\PageIndex{13}\)) like what happens along the San Andreas fault in California. In many places the crust is separating and moving away in opposite directions, or diverging as happened to create the Great Rift Valley of Africa .
Though many earthquakes seemingly occur along plate boundaries, they can occur far away from the edges of plates too. One of the most well-known seismic regions is the New Madrid Seismic Zone located in the Mississippi Valley of the central United States. Investigate the New Madrid Seismic Zone by reading The Mississippi Valley - "Whole Lotta Shakin' Goin' On" . Then return here to continue.
Video : Plate tectonics: Evidence of plate movement (Courtesy of Khan Academy)