23.2: Digestive System Processes and Regulation
Learning Objectives
- Discuss six fundamental activities of the digestive system, giving an example of each
- Compare and contrast the neural and hormonal controls involved in digestion
The digestive system uses mechanical and chemical activities to break food down into absorbable substances during its journey through the digestive system. Table \(\PageIndex{1}\) provides an overview of the basic functions of the digestive organs.
| Organ | Major functions | Other functions |
|---|---|---|
| Mouth |
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| Pharynx |
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| Esophagus |
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| Stomach |
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| Small intestine |
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| Accessory organs |
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| Large intestine |
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Digestive Processes
The processes of digestion include six activities: ingestion, propulsion, mechanical or physical digestion, chemical digestion, absorption, and defecation.
The first of these processes, ingestion , refers to the entry of food into the alimentary canal through the mouth. There, the food is chewed and mixed with saliva, which contains enzymes that begin breaking down the carbohydrates in the food plus some lipid digestion via lingual lipase. Chewing increases the surface area of the food and allows an appropriately sized bolus to be produced.
Food leaves the mouth when the tongue and pharyngeal muscles propel it into the esophagus. This act of swallowing, the last voluntary act until defecation, is an example of propulsion , which refers to the movement of food through the digestive tract. It includes both the voluntary process of swallowing and the involuntary process of peristalsis. Peristalsis consists of sequential, alternating waves of contraction and relaxation of alimentary wall smooth muscles, which act to propel food along (Figure \(\PageIndex{1}\)). These waves also play a role in mixing food with digestive juices.
Digestion includes both mechanical and chemical processes. Mechanical digestion is a purely physical process that does not change the chemical nature of the food. Instead, it makes the food smaller to increase both surface area and mobility. It includes mastication , or chewing, as well as tongue movements that help break food into smaller bits and mix food with saliva. Although there may be a tendency to think that mechanical digestion is limited to the first steps of the digestive process, it occurs after the food leaves the mouth, as well. The mechanical churning of food in the stomach serves to further break it apart and expose more of its surface area to digestive juices, creating an acidic “soup” called chyme . Segmentation , which occurs mainly in the small intestine, consists of localized contractions of circular muscle of the muscularis layer of the alimentary canal. These contractions isolate small sections of the intestine, moving their contents back and forth while continuously subdividing, breaking up, and mixing the contents. By moving food back and forth in the intestinal lumen, segmentation mixes food with digestive juices and facilitates absorption.
In chemical digestion , starting in the mouth, digestive secretions break down complex food molecules into their chemical building blocks (for example, proteins into separate amino acids). These secretions vary in composition, but typically contain water, various enzymes, acids, and salts. The process is completed in the small intestine.
Food that has been broken down is of no value to the body unless it enters the bloodstream and its nutrients are put to work. This occurs through the process of absorption , which takes place primarily within the small intestine. There, most nutrients are absorbed from the lumen of the alimentary canal into the bloodstream through the epithelial cells that make up the mucosa. Lipids are absorbed into lacteals and are transported via the lymphatic vessels to the bloodstream (the subclavian veins near the heart). The details of these processes will be discussed later.
In defecation , the final step in digestion, undigested materials are removed from the body as feces.
AGING AND THE... Digestive System: From Appetite Suppression to Constipation
Age-related changes in the digestive system begin in the mouth and can affect virtually every aspect of the digestive system. Taste buds become less sensitive, so food isn’t as appetizing as it once was. A slice of pizza is a challenge, not a treat, when you have lost teeth, your gums are diseased, and your salivary glands aren’t producing enough saliva. Swallowing can be difficult, and ingested food moves slowly through the alimentary canal because of reduced strength and tone of muscular tissue. Neurosensory feedback is also dampened, slowing the transmission of messages that stimulate the release of enzymes and hormones.
Pathologies that affect the digestive organs—such as hiatal hernia, gastritis, and peptic ulcer disease—can occur at greater frequencies as you age. Problems in the small intestine may include duodenal ulcers, maldigestion, and malabsorption. Problems in the large intestine include hemorrhoids, diverticular disease, and constipation. Conditions that affect the function of accessory organs—and their abilities to deliver pancreatic enzymes and bile to the small intestine—include jaundice, acute pancreatitis, cirrhosis, and gallstones.
In some cases, a single organ is in charge of a digestive process. For example, ingestion occurs only in the mouth and defecation only in the anus. However, most digestive processes involve the interaction of several organs and occur gradually as food moves through the alimentary canal (Figure \(\PageIndex{2}\)).
Some chemical digestion occurs in the mouth. Some absorption can occur in the mouth and stomach, for example, alcohol and aspirin.
Regulatory Mechanisms
Neural and endocrine regulatory mechanisms work to maintain the optimal conditions in the lumen needed for digestion and absorption. These regulatory mechanisms, which stimulate digestive activity through mechanical and chemical activity, are controlled both extrinsically and intrinsically.
Neural Controls
The walls of the alimentary canal contain a variety of sensors that help regulate digestive functions. These include mechanoreceptors, chemoreceptors, and osmoreceptors, which are capable of detecting mechanical, chemical, and osmotic stimuli, respectively. For example, these receptors can sense when the presence of food has caused the stomach to expand, whether food particles have been sufficiently broken down, how much liquid is present, and the type of nutrients in the food (lipids, carbohydrates, and/or proteins). Stimulation of these receptors provokes an appropriate reflex that furthers the process of digestion. This may entail sending a message that activates the glands that secrete digestive juices into the lumen, or it may mean the stimulation of muscles within the alimentary canal, thereby activating peristalsis and segmentation that move food along the intestinal tract.
The walls of the entire alimentary canal are embedded with nerve plexuses that interact with the central nervous system and other nerve plexuses—either within the same digestive organ or in different ones. These interactions prompt several types of reflexes. Extrinsic nerve plexuses orchestrate long reflexes, which involve the central and autonomic nervous systems and work in response to stimuli from outside the digestive system. Short reflexes, on the other hand, are orchestrated by intrinsic nerve plexuses within the alimentary canal wall. These two plexuses and their connections were introduced earlier as the enteric nervous system. Short reflexes regulate activities in one area of the digestive tract and may coordinate local peristaltic movements and stimulate digestive secretions. For example, the sight, smell, and taste of food initiate long reflexes that begin with a sensory neuron delivering a signal to the medulla oblongata. The response to the signal is to stimulate cells in the stomach to begin secreting digestive juices in preparation for incoming food. In contrast, food that distends the stomach initiates short reflexes that cause cells in the stomach wall to increase their secretion of digestive juices.
Hormonal Controls
A variety of hormones are involved in the digestive process. The main digestive hormone of the stomach is gastrin, which is secreted in response to the presence of food. Gastrin stimulates the secretion of gastric acid by the parietal cells of the stomach mucosa. Other GI hormones are produced and act upon the gut and its accessory organs. Hormones produced by the duodenum include secretin, which stimulates a watery secretion of bicarbonate by the pancreas; cholecystokinin (CCK), which stimulates the secretion of pancreatic enzymes and bile from the liver and release of bile from the gallbladder; and gastric inhibitory peptide, which inhibits gastric secretion and slows gastric emptying and motility. These GI hormones are secreted by specialized epithelial cells, called endocrinocytes, located in the mucosal epithelium of the stomach and small intestine. These hormones then enter the bloodstream, through which they can reach their target organs.
Chapter Review
The digestive system ingests and digests food, absorbs released nutrients, and excretes food components that are indigestible. The six activities involved in this process are ingestion, motility, mechanical digestion, chemical digestion, absorption, and defecation. These processes are regulated by neural and hormonal mechanisms.
Interactive Link Questions
Visit this site for an overview of digestion of food in different regions of the digestive tract. Note the route of non-fat nutrients from the small intestine to their release as nutrients to the body.
Answer: Answers may vary.
Multiple Choice
Q. Which of these processes occurs in the mouth?
A. ingestion
B. mechanical digestion
C. chemical digestion
D. all of the above
Answer: D
Q. Which of these processes occurs throughout most of the alimentary canal?
A. ingestion
B. propulsion
C. segmentation
D. absorption
Answer: B
Q. Which of the following stimuli activates sensors in the walls of digestive organs?
A. breakdown products of digestion
B. distension
C. pH of chyme
D. all of the above
Answer: D
Q. Which of these statements about reflexes in the GI tract is false?
A. Short reflexes are provoked by nerves near the GI tract.
B. Short reflexes are mediated by the enteric nervous system.
C. Food that distends the stomach initiates long reflexes.
D. Long reflexes can be provoked by stimuli originating outside the GI tract.
Answer: A
Critical Thinking Questions
Q. Offer a theory to explain why segmentation occurs and peristalsis slows in the small intestine.
A. The majority of digestion and absorption occurs in the small intestine. By slowing the transit of chyme, segmentation and a reduced rate of peristalsis allow time for these processes to occur.
Q. It has been several hours since you last ate. Walking past a bakery, you catch a whiff of freshly baked bread. What type of reflex is triggered, and what is the result?
A. The smell of food initiates long reflexes, which result in the secretion of digestive juices.
Glossary
- absorption
- passage of digested products from the intestinal lumen through mucosal cells and into the bloodstream or lacteals
- chemical digestion
- enzymatic breakdown of food
- chyme
- soupy liquid created when food is mixed with digestive juices
- defecation
- elimination of undigested substances from the body in the form of feces
- ingestion
- taking food into the GI tract through the mouth
- mastication
- chewing
- mechanical digestion
- chewing, mixing, and segmentation that prepares food for chemical digestion
- peristalsis
- muscular contractions and relaxations that propel food through the GI tract
- propulsion
- voluntary process of swallowing and the involuntary process of peristalsis that moves food through the digestive tract
- segmentation
- alternating contractions and relaxations of non-adjacent segments of the intestine that move food forward and backward, breaking it apart and mixing it with digestive juices


21.5: Visuals
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Visuals can be considered to be “real” as in photographs and slides. Visuals can also be pictorial or diagrammatical representations of “real” objects.
For instructional purposes, you can use visuals to illustrate objects and ideas, identify objects, show relationships between objects and ideas, classify objects, show spatial relationships, teach psychomotor skills that do not need to be recognized or copied, and help teach attitudes. You can also use visuals to make abstract concepts concrete. For example, you can do this by graphically showing the relationship between interest rates and the time required to pay off a mortgage. When you design online lessons follow the old adage, “A picture is worth a thousand words.” You should consider combining visuals with text to provide practice and feedback, as text alone does not suffice in many cases. If you teach with visuals, you should likely include visuals in practice and feedback as well as testing.
Note that some learners have difficulty learning from abstract sources such as text, numbers, and symbols. Visuals are an alternative that can help these learners. Similarly, you can help these learners by presenting data with graphs and charts.
Compared to only using text, visuals combined with text reduce the learning time and help students acquire and retain information. Visuals provide an alternate learning path since certain parts of the brain process visuals while different parts process text.
Note
In general, adding visuals leads to increased learning.
Visuals are particularly valuable when you direct learners to focus their attention on specific details, which you can do through concise prompts or captions. A combination of text and visuals can result in a 15 to 50 percent increase in recall over either alone. Recall is increased for both simple skills, such as recalling facts, and complex skills, like performing operations. Lessons with visuals result in higher retention over long time periods (i.e., months). Visuals particularly benefit weak learners. Also, they can be mentally processed much faster than text. Students may be better able to transfer the skills learned to other situations.
In general, visuals can be valuable learning tools. Some ideas cannot be adequately expressed in words but can be readily depicted with visuals. How could you adequately describe the surface of Venus or the structure of DNA without a visual? Look at Figure \(\PageIndex{1}\). Think about the words you would use to describe what this rhinoceros looks like. If the learner has not seen one before, with those words and without seeing a visual, could learners accurately envision the rhinoceros? For online applications, visuals can be particularly effective if students can control the length of time the visual remains on the screen, in other words proceed when they are ready.
Note that in some cases, a visual drawn by an artist may be more effective than a real image. Some learners may not be able to focus on all of the details that real images sometimes provide. In a related way, people will usually recognize a cartoonist’s sketch of a hand as being a hand faster than a digitized image of a hand, even if a cartoonist’s sketch of a hand has three fingers. Simultaneously look at Figures \(\PageIndex{2}\) and \(\PageIndex{3}\). Which do you more quickly recognize as a hand? Since complex visuals require more time to mentally process than simple images, provide simple images (where appropriate).
On screens with text and visuals, learners are naturally drawn to the visuals. As a minimum, your visuals will add variety to screens. Although irrelevant visuals do not help students learn relevant content, some images (e.g., humorous) can help “lighten” the material. Screen variety can increase attention and motivation and consequently indirectly affect learning positively.
Keeping the Message of Your Visual Clear
To keep the message clear, your visuals should be self-explanatory, have labels, and only include relevant information!
Visuals should be self-explanatory
Self-explanatory visuals illustrate the message. If the image is not self-explanatory then you should determine how it could be done. At times, this goal may not be possible but you should at least aim for it. However, be sure to provide an explanation to link the visual to the idea you are conveying.
Note
As a rule, ensure your visuals are self-explanatory.
You should use the following methods to ensure message clarity:
- Keep the visual simple and only include essential information. For example, crop images to eliminate unnecessary details. Complex visuals can weaken the instruction by confusing or distracting the students.
- Match the visual’s complexity to the learner’s skill. Overly complex visuals are often ignored.
- Consider using simple diagrams instead of realistic images. Realistic images sometimes contain so much detail that the learner is distracted from understanding the message.
- Use a series of diagrams rather than a single complicated diagram. When explaining a process, use one visual per step. One technique you can use is to gradually build the visual from screen to screen.
- Add labels to diagrams to highlight key points. Horizontal labels are the easiest to read.
Only include information relevant to the visual
If you include extra information, you could clutter the visual and can cause confusion. On screens that build, some of the previous information may need to be erased or de-highlighted so that new key points are easily found. Build the visuals in logical straightforward sequential steps. Plan to allow the learner the capability to reverse the steps for easy review.


11.13: Avro Pärt
Arvo Pärt enjoys enormous popularity for a “classical” composer. His works are often categorized as holy minimalism though he does not use that term when speaking of his own music. Notice the impact and influence in his personal history of composers and styles we have already studied: early 20th century techniques (including twelve-tone), Soviet artistic repression, minimalism, Medieval chant, and Renaissance polyphony.
Introduction
Arvo Pärt (born 11 September 1935) is an Estonian composer of classical and sacred music. Since the late 1970s, Pärt has worked in a minimalist style that employs his self-invented compositional technique, tintinnabuli. His music is in part inspired by Gregorian chant.
Compositions
Pärt’s works are generally divided into two periods. He composed his early works using a range of neo-classical styles influenced by Shostakovich, Prokofiev, and Bartók. He then began to compose using Schoenberg’s twelve-tone technique and serialism. This, however, not only earned the ire of the Soviet establishment, but also proved to be a creative dead-end. When early works were banned by Soviet censors, Pärt entered the first of several periods of contemplative silence, during which he studied choral music from the 14th to 16th centuries. In this context, Pärt’s biographer, Paul Hillier, observed that “He had reached a position of complete despair in which the composition of music appeared to be the most futile of gestures, and he lacked the musical faith and willpower to write even a single note.”
The spirit of early European polyphony informed the composition of Pärt’s transitional Third Symphony (1971); thereafter he immersed himself in early music, reinvestigating the roots of Western music. He studied plainsong, Gregorian chant and the emergence of polyphony in the European Renaissance.
The music that began to emerge after this period was radically different. This period of new compositions included Fratres , Cantus In Memoriam Benjamin Britten and Tabula Rasa. Pärt describes the music of this period as tintinnabuli—like the ringing of bells. Spiegel im Spiegel (1978) is a well-known example which has been used in many films. The music is characterised by simple harmonies, often single unadorned notes, or triads, which form the basis of Western harmony. These are reminiscent of ringing bells. Tintinnabuli works are rhythmically simple and do not change tempo. Another characteristic of Pärt’s later works is that they are frequently settings for sacred texts, although he mostly chooses Latin or the Church Slavonic language used in Orthodox liturgy instead of his native Estonian language. Large-scale works inspired by religious texts include St. John Passion , Te Deum , and Litany . Choral works from this period include Magnificat and The Beatitudes .
Of Pärt’s popularity, Steve Reich has written: “Even in Estonia, Arvo was getting the same feeling that we were all getting. . . .  He’s completely out of step with the zeitgeist and yet he’s enormously popular, which is so inspiring. His music fulfills a deep human need that has nothing to do with fashion.” Pärt’s music came to public attention in the West largely thanks to Manfred Eicher who recorded several of Pärt’s compositions for ECM Records starting in 1984.
Invited by Walter Fink, Pärt was the 15th composer featured in the annual Komponistenporträt of the Rheingau Musik Festival in 2005 in four concerts. Chamber music included Für Alina for piano, played by himself, Spiegel im Spiegel and Psalom for string quartet. The chamber orchestra of the Bavarian Radio Symphony Orchestra played his Trisagion , Fratres and Cantus along with works of J.S. Bach. The Windsbach Boys Choir and soloists Sibylla Rubens, Ingeborg Danz, Markus Schäfer and Klaus Mertens performed Magnificat and Collage über B-A-C-H together with two cantatas of Bach and one of Mendelssohn. The Hilliard Ensemble, organist Christopher Bowers-Broadbent, the Rostock Motet Choir and the Hilliard instrumental ensemble, conducted by Markus Johannes Langer, performed a program of Pärt’s organ music and works for voices (some a cappella ), including Pari Intervallo , De profundis , and Miserere .
A new composition, Für Lennart , written for the memory of the Estonian President, Lennart Meri, was played at Meri’s funeral service on 2 April 2006.
In response to the murder of the Russian investigative journalist Anna Politkovskaya in Moscow on 7 October 2006, Pärt declared that all of his works performed in 2006 and 2007 would be in honour of her death, issuing the following statement: “Anna Politkovskaya staked her entire talent, energy and—in the end—even her life on saving people who had become victims of the abuses prevailing in Russia.”
Pärt was honoured as the featured composer of the 2008 RTÉ Living Music Festival in Dublin, Ireland. He was also commissioned by Louth Contemporary Music Society to compose a new choral work based on “St. Patrick’s Breastplate,” which premiered in 2008 in Louth, Ireland. The new work is called The Deers Cry . This is his first Irish commission, having its debut in Drogheda and Dundalk in February 2008.
Pärt’s 2008 Symphony No. 4 is named “Los Angeles” and was dedicated to Mikhail Khodorkovsky. It was Pärt’s first symphony written since his Symphony No. 3 written in 1971. It premiered in Los Angeles, California, at the Walt Disney Concert Hall on 10 January 2009, and has been nominated for a GRAMMY for Best Classical Contemporary Composition.
On 10 December 2011, Pärt was appointed a member of the Pontifical Council for Culture for a five-year renewable term by Pope Benedict XVI.
On 26 January 2014, Pärt’s Adam’s Lament won a Grammy for Best Choral Performance.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Arvo Part. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Arvo_P%C3%A4rt . License : CC BY-SA: Attribution-ShareAlike


4.2A: Components of Plasma Membranes
- Describe the function and components of the plasma membrane
Structure of Plasma Membranes
The plasma membrane (also known as the cell membrane or cytoplasmic membrane) is a biological membrane that separates the interior of a cell from its outside environment.
The primary function of the plasma membrane is to protect the cell from its surroundings. Composed of a phospholipid bilayer with embedded proteins, the plasma membrane is selectively permeable to ions and organic molecules and regulates the movement of substances in and out of cells. Plasma membranes must be very flexible in order to allow certain cells, such as red blood cells and white blood cells, to change shape as they pass through narrow capillaries.
The plasma membrane also plays a role in anchoring the cytoskeleton to provide shape to the cell, and in attaching to the extracellular matrix and other cells to help group cells together to form tissues. The membrane also maintains the cell potential.
In short, if the cell is represented by a castle, the plasma membrane is the wall that provides structure for the buildings inside the wall, regulates which people leave and enter the castle, and conveys messages to and from neighboring castles. Just as a hole in the wall can be a disaster for the castle, a rupture in the plasma membrane causes the cell to lyse and die.
The Plasma Membrane and Cellular Transport
The movement of a substance across the selectively permeable plasma membrane can be either “passive”—i.e., occurring without the input of cellular energy —or “active”—i.e., its transport requires the cell to expend energy.
The cell employs a number of transport mechanisms that involve biological membranes:
- Passive osmosis and diffusion: transports gases (such as O 2 and CO 2) and other small molecules and ions
- Transmembrane protein channels and transporters: transports small organic molecules such as sugars or amino acids
- Endocytosis: transports large molecules (or even whole cells) by engulfing them
- Exocytosis: removes or secretes substances such as hormones or enzymes
The Plasma Membrane and Cellular Signaling
Among the most sophisticated functions of the plasma membrane is its ability to transmit signals via complex proteins. These proteins can be receptors, which work as receivers of extracellular inputs and as activators of intracellular processes, or markers, which allow cells to recognize each other.
Membrane receptors provide extracellular attachment sites for effectors like hormones and growth factors, which then trigger intracellular responses. Some viruses, such as Human Immunodeficiency Virus (HIV), can hijack these receptors to gain entry into the cells, causing infections.
Membrane markers allow cells to recognize one another, which is vital for cellular signaling processes that influence tissue and organ formation during early development. This marking function also plays a later role in the “self”-versus-“non-self” distinction of the immune response. Marker proteins on human red blood cells, for example, determine blood type (A, B, AB, or O).
LICENSES AND ATTRIBUTIONS
Key Points
- The principal components of the plasma membrane are lipids ( phospholipids and cholesterol), proteins, and carbohydrates.
- The plasma membrane protects intracellular components from the extracellular environment.
- The plasma membrane mediates cellular processes by regulating the materials that enter and exit the cell.
- The plasma membrane carries markers that allow cells to recognize one another and can transmit signals to other cells via receptors.
Key Terms
- plasma membrane : The semipermeable barrier that surrounds the cytoplasm of a cell.
- receptor : A protein on a cell wall that binds with specific molecules so that they can be absorbed into the cell.


15.26: Discussion- Academic Honesty
STEP 1 : Click on the link and read all of the corresponding background information about it from The New York Times, including the questions at the bottom of the page. Note that this article is now quite dated. Do you think it’s still relevant? In your opinion, has cheating gotten worse or better since 2012? In what ways?
STEP 2 : Post your reply to the question above (in at least 250 words) in the discussion forum. Also address similar questions, such as: What do you think should be done about cheating or academic honesty? Do you think your peers understand what plagiarism means? Do you think your instructors are aware of the cheating that takes place on your campus?
Your response should include a clear stance on the issue with supporting evidence as to why you feel the way you do. Show that you can think critically on the topic by integrating your own thoughts, analysis, or experiences.
STEP 3 : Respond in two separate posts to two classmates (in at least 75 words). Explicitly address their responses and try to extend, complicate, or redirect their points in a substantive, knowledge-demonstrating way.
Contributors and Attributions
- Provided by : Lumen Learning. License : CC BY: Attribution


1.12: The Definite Article and Persistent Accent
Module 11
© 2021 Philip S. Peek, CC BY 4.0 https://doi.org/ 10.11647/OBP.0264. 12
The Definite Article and Adjectives
In Greek and in English the definite article, the , is an adjective. Greek and English adjectives have the same function, though the Greek adjective has endings and the English adjective does not. Adjectives in both languages are words that describe nouns and pronouns. In the below,
a midnight dreary ,
and
I ponder, weak and weary ,
a and dreary describe the noun midnight ; weak and weary describe the pronoun, I . In English, words can change their function and part of speech without changing form. Consider the sentence,
I left my gal and home .
In this sentence home is a noun. In the sentence,
I go home ,
home is an adverb. In the sentences,
I miss my home town,
and
I hit a home run,
home is an adjective. In Greek, words do not typically change function and part of speech without changing form. In Greek the adjective has endings becauses each adjective agrees in gender, case, and number with the noun it modifies.
The Definite Article
Though in Greek and in English the definite article is an adjective, the functions of the two definite articles have important similarities and differences. Often the Greek definite article, ὁ , ἡ , τό , is best translated with its English equivalent the . In this case the function of the article in both languages is the same. In other situations English calls for a translation of the Greek article with a possessive adjective: my, your, his, her, our, your, their . As you proceed through this text and Part II of the 21st-Century series, you will encounter additional differences between the way the Greek and English articles function and the meanings they create.
Memorize the forms of the article. Be sure to memorize letters, breathings, and accents so that you can reproduce from memory each form of the article.
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- Note that the masculine and feminine singular and plural nominative forms, ὁ , ἡ , οἱ , and αἱ do not have an accent. They are proclitics and pronounced so closely with the word that follows them that they almost form a single word. All of the genitive and dative singular and plural forms have a circumflex accent. The remaining nominative and accusative forms have an acute accent.
Remember that the article is an adjective . In Greek and in English adjectives are words that describe nouns. In Greek, adjectives agree with the nouns that they modify in gender, number, and case. In translating Greek, the ability to see the relationship between an adjective and the noun it modifies is one that you will use in every sentence you translate. Consequently, understanding how the adjective relates to the noun it modifies is essential to your learning the language. The article is the most commonly occurring adjective in ancient Greek. It is also very versatile. In the next module, we will see how it affects meaning in three common instances.
Persistent Accent
Adjectives, adverbs, conjunctions, nouns, and pronouns mainly have a persistent accent. Since they are declined, accents on adjectives, nouns, and pronouns can change nature—acute, grave, circumflex—and position—antepenult, penult, ultima. When presented with any noun or pronoun in a lexicon, the nominative case of the noun or pronoun is given first, the genitive case of the noun or pronoun, second, and the article, third:
|
Nominative Singular |
Genitive Singular |
Article |
English Equivalent |
|
αἷμα |
αἵματος |
τό |
blood |
Remember that the article indicates gender. Persistent accent remains the same accent (acute, grave, circumflex) over the vowel or diphthong it is on, as given by the nominative singular in all forms of the word, unless forced by the possibilities of accent to change in nature (acute, circumflex, grave) or position (antepenult, penult, ultima). If an accent violates one of the possibilities (you cannot have a circumflex on the antepenult), the accent will change in nature (acute, grave, circumflex) before position (antepenult, penult, ultima). The accent of most noun forms is persistent and is learned as part of the vocabulary.
Vowel Length
In the paradigms and accenting practice of this text, macrons mark alpha, iota, and upsilon if long. Short vowels and diphthongs are not marked. In authentic texts and in the Practice Translating of this text, macrons do not occur. Diphthongs are by definition long with this exception: final - αι and - οι are short for purposes of accentuation except in the optative, a mood learned in Part II of the 21st-Century series. - αι and - οι are final when they appear as the last two letters of a word, λῦσ αι but not λύσ αις .
Review Possibilities of Accent
Review these four possibilities, rememorizing them if you need to.
Acute Accent
- An acute accent can appear on the antepenult, penult, or ultima.
- An acute accent can only appear on the antepenult if the ultima is short.
Circumflex Accent
- A circumflex accent can appear only on long vowels and never accents the antepenult.
- A circumflex accent can appear on the penult if the penult is long and the ultima is short. Try using the mnemonic PLUS: Penult Long Ultima Short.
Additional Possibilities of Accent
Memorize these additional possibilities.
Acute Accent
- An acute accent can appear on the antepenult, penult, or ultima.
- An acute accent can only appear on the antepenult if the ultima is short.
- An acute accent on the ultima changes to a grave when followed by a second word in a sentence.
- An acute accent on the ultima does not change to a grave when followed by a second word in a sentence if there is a pause (comma, raised dot, period, question mark).
Circumflex Accent
- A circumflex accent can appear only on long vowels and never accents the antepenult.
- A circumflex accent can appear on the penult if the penult is long and the ultima is short. Remember PLUS: Penult Long Ultima Short.
- A circumflex accent can appear on a long ultima.
Grave Accent
- A grave accent can only appear on the ultima.
- A grave accent can only appear on the ultima when a second word in a sentence follows without a pause. A pause is indicated by a comma, raised dot, period, or question mark.
Chart for Possibilities of Accent
Key: a stands for antepenult; pe for penult; and u for ultima.
|
Acute |
Grave |
Circumflex |
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Antepenult |
Possible if ultima is short: á-pe-ŭ |
Never |
Never |
|
Penult |
Possible but not if penult is long and ultima is short: a-pé-u |
Never |
Possible if penult is long and ultima is short: a-pê-ŭ |
|
Ultima |
Possible if pause follows: a-pe-ú + pause between words |
Possible if no pause follows: a-pe-ù + no pause between words |
Possible: a-pe-ũ |
If, when reading the above, your head is left spinning, do not worry. Most people need to start applying what they are learning before they can begin to enter into understanding. In the below exercises you can learn from the examples. As you need to, go back and reread the possibilities for accent, committing them to memory through applied practice.
Application of the Chart for Possibilities of Accent in Persistent Accent. Consider the following examples.
|
1. |
ἄνθρωπος (nominative): |
ἀνθρώπου , ἀνθρώπῳ |
|
2. |
βιβλίον (nominative): |
βιβλίου , βιβλίῳ |
|
3. |
νῆσος (nominative): |
νήσου , νήσῳ , νῆσον |
|
4. |
δρᾶμα (nominative): |
δράματος , δραμά των |
|
5. |
ἀρετή (nominative): |
ἀρετήν , ἀρετάς |
Explanations.
- ἀνθρώπου , ἀνθρώπῳ : the ultima is long and so the acute accent must change in position from the antepenult to the penult, but not in nature.
- βιβλίου , βιβλίῳ : no violation of the possibilities and so no change.
- νήσου , νήσῳ : the penult and ultima are long and so the accent must change in nature from a circumflex to an acute, but need not change position. νῆσον : the penult is long and the ultima is short and so the accent remains a circumflex on the penult.
- δράματος : the accent remains over the syllable δρᾱ but must change in nature to an acute, because the number of syllables changed from two to three and it is not possible to have a circumflex on the antepenult. δρᾱμάτων : the accent must change position because the ultima is long.
- ἀρετήν , ἀρετάς : there is no violation of the possibilities and so no change.
Use these examples and the Chart on Possibilities of Accent to help you complete the persistent accent practice.
Practice with Persistent Accent. The first word in bold gives the persistent accent. Accent the unbolded words. Check your answers with those in the Answer Key .
- θεός : θεον , θεοι , θεους
- λόγος : λογου , λογῳ , λογον , λογων , λογοις , λογους , λογοι
- ἄνθρωπος : ἀνθρωπου , ἀνθρωπῳ , ἀνθρωπον , ἀνθρωπων , ἀ νθρωποις , ἀνθρωποι
- ἀρχή : ἀρχην , ἀρχαι , ἀρχᾱς
- πρᾶγμα : πραγματος , πραγματι , πραγματα , πραγματων
- πόλεμος : πολεμου , πολεμῳ , πολεμον , πολεμων , πολεμοις , πολεμους , πολεμοι
- λιμήν : λιμενος , λιμενι , λιμενα , λιμενες , λιμενων , λιμενας
- χώρᾱ : χωρᾱς , χωραις , χωραι
- ὄνομα : ὀνοματος , ὀνοματι , ὀνοματα , ὀνοματων
- σῶμα : σωματος , σωματι , σωματα , σωματων
There is additional accent practice in Appendix X.
Practice Translating . Translate the sentences below adapted from Euripides’ Alkestis ( Ἄλκηστις ). Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded; genitive endings are italicized; dative endings are highlighted; and accusative endings are underlined. Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
|
Case |
Ending |
Function |
|
Nominative |
- α , - αι , - ευς , - η , - ηρ , - ο , - ος |
subject of the verb |
|
Genitive |
- ας , - ης , - ος , - ου , - ους , - ων |
possession; dependence; object of a verb or preposition |
|
Dative |
- ῃ , - ι , - οις , - ῳ |
indirect object of the verb; object of a preposition |
|
Accusative |
- α , - αν , - ας , - ε , - ην , - ον , - ος , - ους |
object of the verb, preposition, or prefix |
Ἀπόλλων: ὁ Ζ εὺς κτείνει παῖδ α τ ὸν ἐμ όν , Ἀσκληπι όν , καὶ στέρν οις ἐμβάλλει φλόγ α . αὐτ ὸν γὰρ χολόω ἐπεὶ τ οὺς τέκτον ας Δί ου πυρ ὸς κτείνω , τ οὺς Κύκλωπ ας . θητεύειν δὲ ἐμ ὲ ὁ πατ ὴρ θνητ ῷ παρ ’ ἀνδρ ὶ τούτ ων ἄποιν α ἀναγκάζει . ἔρχομαι δὲ τ ὴν εἰς γαῖ αν καὶ βουφορβέω ξέν ῳ καὶ τ ὸν σῴζω οἶκ ον . ὅσι ὸς μέν εἰμι καὶ ὁσί ου δὲ ἀνδρ ὸς τυγχάνω , παιδ ὸς Φέρητ ος , Ἀδμήτ ου · αὐτ ὸν θανεῖν ῥύομαι καὶ Μοίρ ᾱς δολόω . αἰνέουσι δὲ ἐμο ὶ αἱ θε α ί τε καὶ λέγουσιν ὅτι ὁ δὲ Ἄδμητ ος Ἅιδ ην αυτίκ ’ ἐκφεύγει καὶ διαλλάσσει ἄλλ ον τ οῖς κάτω νεκρ όν . ὁ δὲ πάντ ας δ ’ ἐλέγχει καὶ διεξέρχεται φίλ ους , πατέρ α τε καὶ μητέρ α . ὁ δ ’ οὐχ εὑρίσκει οὐδέν α , πλὴν γυναικ ός · ἡ μήτ ηρ αὐτ ὸν τίκτει ἀλλὰ οὐ ἐθέλει θανεῖν πρὸ αὐτ οῦ . ἡ μὲν γυν ὴ ἐθέλει μηκέτ ’ εἰσοράειν φά ος · ὁ δ ’ αὐτ ὴν κατ ’ οἴκ ους ἐν χερσ ὶ βαστάζει καὶ ἡ δὲ ψυχορραγέει .
Adverbs, Prepositions, and Verbs
|
αἰνέω tell of; praise; promise, vow |
* ἔρχομαι come, go |
|
ἀναγκάζω compel, force |
* ἔχω have, hold; be able + inf.; καλῶς ἔχειν to be well |
|
αὐτίκα immediately |
* θανεῖν to die |
|
βαστάζω lift, raise |
θητεύω be a slave, to serve + dat. |
|
* βούλομαι want, prefer; wish, be willing |
κτείνω kill |
|
βουφορβέω be a cowherd |
λέγω say, tell, speak |
|
διαλλά σσω change, exchange, give in exchange |
μηκέτι no longer |
|
διεξέρχομαι go through; be past, be gone by |
οὐκέτι no more, no longer, no further |
|
δολόω trick, deceive |
πλήν except for + gen. |
|
* εἰμί be, be possible |
ῥύομαι draw to oneself, save ; αὐτὸν θανεῖν ρύομαι I save him from dying |
|
εἰσοράειν to look upon |
σῴζω save |
|
ἐκφεύγω flee, escape |
τίκτω bear, give birth |
|
ἐλέγχω cross-examine; put to the test; prove; refute |
* τυγχάνω obtain, meet + gen. |
|
* εὑρίσκω find |
χολόω anger, upset |
|
* ἐθέλω ( θέλω ) wish, be willing |
ψυχορραγέω let the soul break loose, be at the last breath |
|
ἐμβάλλω throw in, put in; put ‘x’ in acc. into ‘y’ in dat. |
Adjectives, Nouns, Pronouns
|
Νοminative |
Genitive |
Dative |
Accusative |
English Equivalent |
|
Ἄδμητος |
Ἀδμήτου |
Ἀδμήτῳ |
Ἄδμητον |
Admetos |
|
Ἅιδης |
Ἅιδου |
Ἅιδῃ |
Ἅιδην |
Hades |
|
* ἄλλος |
ἄλλου |
ἄλλ ῳ |
ἄλλον |
another, other |
|
* ἀνήρ |
ἀνδρός |
ἀνδρί |
ἄνδρα |
man, husband |
|
ἄποινα |
ἀποίνων |
ἀποίνοις |
ἄποινα |
ransom, payment |
|
Ἀσκληπιός |
Ἀσκληπιοῦ |
Ἀσκληπι ῷ |
Ἀσκληπιόν |
Asklepios |
|
* αὐτή |
αὐτῆς |
αὐτῇ |
αὐτήν |
she, her, hers |
|
* αὐτός |
αὐτοῦ |
αὐτῷ |
αὐτόν |
he, him, his |
|
βουλεύματα |
βουλευμάτων |
βουλεύμασι |
βουλεύματα |
will |
|
βροτοί |
βροτῶν |
βροτοῖς |
βροτούς |
mortals |
|
γαῖα |
γαίας |
γαίᾳ |
γαῖαν |
earth, land |
|
* γυνή |
γυναικός |
γυναικί |
γυναῖκα |
woman, wife |
|
* ἐγώ |
ἐμοῦ |
ἐμοί |
ἐμέ |
I, me, mine |
|
* ἐμόν |
ἐμοῦ |
ἐμῷ |
ἐμόν |
my |
|
* Ζεύς |
Διός |
Διί |
Δία |
Zeus |
|
* θεαί |
θεῶν |
θεαῖς |
θεάς |
goddesses |
|
θνητός |
θνητοῦ |
θνητῷ |
θνητόν |
mortal |
|
Κύκλωπες |
Κυκλ ώπων |
Κύκλωψι ( ν ) |
Κύκλωπας |
Kyklopes |
|
* μήτηρ |
μητρό ς |
μητρί |
μητέρα |
mother |
|
Μοῖραι |
Μοιρῶν |
Μοίραις |
Μοίρας |
Fates |
|
νεκρός |
νεκροῦ |
νεκρῷ |
νεκρόν |
corpse |
|
ξένος ( ξεῖνος ) |
ξένου |
ξένῳ |
ξένον |
stranger, guest-friend |
|
οἱ κάτω |
τῶν κάτω |
τοῖς κάτω |
τοὺς κάτω |
those below |
|
οἶκος |
οἴκου |
ο ἴκῳ |
οἶκον |
house, palace |
|
ὅσιος |
ὁσίου |
ὁ σίῳ |
ὅσιον |
devout, holy |
|
* οὐδείς |
οὐδένος |
οὐδένι |
οὐδένα |
noone |
|
* παῖς |
παιδός |
παιδί |
παῖδα |
child |
|
* πάντες |
πάντων |
πάσι ( ν ) |
πάντας |
all |
|
* πατήρ |
πατρός |
πατρί |
πατέρα |
father |
|
πῦρ |
πυρός |
πυρί |
πῦρ |
fire |
|
στέρνον |
στέρνου |
στέρνῳ |
στέρνον |
breast, chest |
|
* ταῦτα |
τούτων |
τούτοις |
ταῦτα |
these things |
|
τέκτονες |
τεκτόνων |
τέκτοσι ( ν ) |
τέκτονας |
makers, artisans |
|
φάος ( φῶς ) |
φάεος ( φωτός ) |
φάει ( φωτί ) |
φάος ( φῶς ) |
light, daylight |
|
Φέρης |
Φέρητος |
Φέρητι |
Φέρητα |
Pheres |
|
* φίλος |
φίλου |
φίλῳ |
φίλον |
friend, loved-one |
|
φλόξ |
φλογός |
φλογί |
φλόγα |
flame, fire |
|
* χείρ |
χειρός |
χειρί |
χεῖρα |
hand; force, army |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Parsing Greek Sentences. Parse each word of the sentence found below. For nouns and pronouns, give case and function. For verbs, give person, number, tense, mood, and voice. For adverbs and conjunctions, identify them. For prepositional phrases, give the preposition and the preposition’s object. For adjectives, tell what noun they agree with in gender, number, and case.
ὁ Ζεὺς κτείνει παῖδα τὸν ἐμόν , Ἀσκληπιόν , καὶ στέρνοις ἐμβ άλλει φλόγα .
Check your answers with those in the Answer Key .
Herodotos
Born in Halikarnessos (Halicarnassus), a gateway between the Greek and Persian worlds, Herodotos began the telling of history. Most biographical information about him is gleaned from his historical work, his Histories . In writing it, Herodotos traveled throughout the Mediterranean and the surrounding lands, interviewing sources and looking over data. In helping to create the genre of history, Herodotos focuses on the eighty-two years from 560 to 478 BCE. Outside of this frame he looks back to the origins of the first eight gods, and forward to the events of the Peloponnesian War (431–404 B.C.E.). In crafting his narrative, he makes use of a variety of source material, records events he does and does not believe, and passes judgment where he deems it appropriate, taking pains to establish the truth of what occurred. Where this is not possible, he still preserves what his sources relate. As he worked, he made mistakes and he got things right. His scope is a broad one. As he crafts his narrative, he discusses many other subjects, including the great struggle between the Persian Empire and the Greek-speaking city-states at the dawn of the classical era. These other things include chronology, colonies, customs, deeds, flora and fauna, food, funeral practices, genealogy, geography, great works, lineage, marriage, origins, religion, and sex. Herodotos does not merely list events or tell tales; his history inquires into the causes of events, and casts its net wide to include ethnography and legend as well as military and political history. Though parts of his work read as though Herodotos is a carnival barker, calling his audience to view the strange and incredible world of flying snakes, fish-eating horses, and gold-digging ants, underlying even these fantastic accounts is a reasonable and rational mind, seeking to present what it has gone to great lengths to discover. In presenting his findings, Herodotos weaves a variety of perspectives into his work, creating a complexity that recognizes the difficulties involved in explaining the past. The result is a rich narrative, full of nuance, that offers certainty when it can and uncertainty when it cannot.
Module 11 Top 250 Vocabulary to be Memorized. Like learning the alphabet and endings, memorizing vocabulary is essential to acquiring language. The better you memorize the top 250 most frequently occurring words, the greater mastery of the language you will have.
Adjective and Particle
ὁ , ἡ , τό the (proclitic, ὁ , ἡ , οἱ , αἱ )
ὦ (precedes a noun, marks for the vocative case, often not translated) oh
Verbs
ἄγω do, drive, lead
ἀκούω hear, hear of or about, listen, heed + gen. or acc. of thing or gen. of person
ἄρχω rule, command; begin + gen.
δύναμαι be able, be strong enough + inf.
καλέω call
οἴομαι ( οἶμαι ) think, suppose, believe
πράσσω ( πράττω ) do, make; fare; + κακῶς suffer
φέρω bring, bear, carry; endure


1.9: Training and Development
- Identify human resource management training options
- Identify human resource management development options
Training Options
As the Bureau of Labor Statistics site notes, most human resource positions require a bachelor’s degree. Each listing has a “How to Become a…” section that specifies typical education and related work experience. For example, BLS states that human resource manager candidates need a bachelor’s degree and several years of related work experience at a minimum, with some jobs requiring a master’s degree. Relevant degrees include human resources as well as finance, business management, education, or information technology. At a master’s level, relevant degrees include human resources and business administration (MBA). Relevant courses for human resource roles in general include business communications, finance, human resources management and management. If available, courses in compensation analysis and benefits administration are also of value.
Training for human resource management might include experience in a human resources role—for example, as a human resources specialist. Management positions typically require a knowledge of federal, state and local employment laws, experience administering human resources plans and programs and familiarity with human resources software. In addition to formal education, all human resources roles require strong interpersonal skills. Management roles include additional skills, including (depending on the role), business management, communication, critical thinking, decision-making, leadership and organization.
For additional HR-related training options, refer to HumanresourcesMBA.net’s (and similar) rankings of Human Resource and Organizational Development courses and degree programs at the bottom of their list of the 15 Best HR Organizations and Associations . [1]
Development Options
Engaging in ongoing learning and development is an expectation in every profession, and human resources is no exception. One of the ways to do this is to pursue (and maintain) a professional certification such as the Society of Human Resource Management’s Certified Professional (SHRM-CP) or a specific field of practice credential such as SHRM’s Talent Acquisition or People Analytics Specialty Credential. Professional credentials prove expertise, add to one’s professional credibility and demonstrate commitment. It’s common to see credentials listed as required or desired qualifications on job postings and they are also a differentiator that may be factored in to promotion decisions.
The following is a short-list of formal human resource development options:
- Society for Human Resource Management (SHRM) . The largest HR organization in the world, SHRM positions itself as “the foremost expert, convener and thought leader on issues impacting today’s evolving workplaces.” SHRM produces industry research and publications, offers certification and organizes conferences. SHRM also has a HR People + Strategy group for HR executives with at least 10 years of experience. An affiliate, the Council for Global Immigration, is for HR professionals specializing in employment-based immigration.
- WorldatWork . WorldatWork is professional association for HR professionals that deal with compensation broadly including compensation, executive compensation, sales compensation, benefits, and work life issues. WorldatWork offers certification programs in each of these areas—for example, Certified Compensation Professional (CCP), Certified Sales Compensation Professional (CSCP), and Certified Benefits Professional (CBP). WorldatWork also produces industry research and publications, offers training certification and organizes conferences.
-
HR Certification Institute
. HRCI offers certification options for those who are just starting their HR career and HR experts, with eight credentials reflecting different levels of career advancement and geographic areas of expertise. For example, the Professional in Human Resources certificate is available in California and International options (PHRca and PHRi). To cement the case for certification, HRCI cites a report by the Human Resources Research Organization (HumRRO) that found that “certified HR professionals, specifically those who have earned either the PHR or SPHR, are more likely than non-certified professionals to:
- Get hired and be employed full-time, be well compensated, and be happier with their career.
- Perform better on the job, show greater potential for future positions, perform better on strategic HR efforts, and offer greater HR expertise.
For those interested in the business case for certification and/or evaluating certification options, the study is worth reading in full: [2]
In addition to participating in association meetings and conferences and attending classes, webinars and other structured training, you can set up Google Alerts on topics of interest, join LinkedIn groups, and read/subscribe to/follow relevant blogs, podcasts and Twitter feeds. For an overview of how to find relevant thought leaders, read Undercover Recruiter’s article How to Find People to Follow on Twitter (also provides tips on finding relevant content creators on the web in general). [3] For an initial Twitter list, refer to HR Dive’s 10 must-follow HR Twitter accounts . [4]
- " 15 Best Organizations and Associations for Human Resource Professionals. " Human Resources MBA. Accessed July 29, 2019. ↵
- "↵
- Morgan, Hannah. " 10 Ways to Find Relevant People on Twitter ." Undercover Recruiter. Accessed July 29, 2019. ↵
- Moody, Kathryn. " 10 Must-Follow HR Twitter Accounts (and Chats). " HR Drive. Accessed July 29, 2019. ↵
- Auditorium. Authored by : Mikael Kristenson. Provided by : Unsplash. Located at : https://unsplash.com/photos/3aVlWP-7bg8 . License : CC0: No Rights Reserved . License Terms : Unsplash License
- Untitled. Authored by : rawpixel. Provided by : Pixabay. Located at : pixabay.com/photos/boy-break-browsing-casual-computer-1986107/. License : CC0: No Rights Reserved . License Terms : Pixabay License


3.7: Trisubstituted Benzenes - Additivity of Effects
After completing this section, you should be able to
- predict the position or positions at which electrophilic substitution will occur when a third substituent is introduced into a disubstituted benzene ring.
- explain the observed substitution pattern when a third substituent is introduced into a disubstituted benzene ring.
Orientational Interaction of Substituents
When a benzene ring has two substituent groups, each exerts an influence on subsequent substitution reactions. The activation or deactivation of the ring can be predicted more or less by the sum of the individual effects of these substituents. The site at which a new substituent is introduced depends on the orientation of the existing groups and their individual directing effects. We can identify two general behavior categories, as shown in the following table. Thus, the groups may be oriented in such a manner that their directing influences act in concert, reinforcing the outcome; or are opposed (antagonistic) to each other. Note that the orientations in each category change depending on whether the groups have similar or opposite individual directing effects.
| Antagonistic or Non-Cooperative | Reinforcing or Cooperative |
|
|
|
|
D = Electron Donating Group (ortho/para-directing)
W = Electron Withdrawing Group (meta-directing) |
Reinforcing or Cooperative Substitutions
The products from substitution reactions of compounds with a reinforcing orientation of substituents are easier to predict than those with antagonistic substituents. In p -nitrotoluene, for example, both the methyl and the nitro group direct further substitution to the same position (ortho to the methyl = meta to the nitro). A single product is thus formed on electrophilic substitution.
For example, the six equations shown below are all examples of reinforcing or cooperative directing effects operating in the expected manner. Symmetry, as in the first two cases, makes it easy to predict the site at which substitution is likely to occur. Note that if two different sites are favored, substitution will usually occur at the one that is least hindered by ortho groups.
The first three examples have two similar directing groups in a meta-relationship to each other. In examples 4 through 6, oppositely directing groups have an ortho or para-relationship. The major products of electrophilic substitution, as shown, are the sum of the individual group effects. The strongly activating hydroxyl (–OH) and amino (–NH 2 ) substituents favor dihalogenation in examples 5 and 6.
Spatial Limitation
Further substitution rarely occurs between the two groups in a meta-disubstituted compound because this site is too hindered. Aromatic rings with three adjacent substituents must therefore be prepared by some other route, such as by substitution of an ortho-disubstituted compound.
What product would you expect from bromination of p -methylbenzoic acid?
Strategy
Identify the two substituents present on the ring, decide the directing effect of each and, if necessary, decide which substituent is the stronger activator. In the present case, the carboxyl group (–CO 2 H) is a meta director and the methyl group is an ortho and para director. Both groups direct bromination to the position next to the methyl group, yielding 3-bromo-4-methylbenzoic acid.
Solution
Antagonistic or Non-Cooperative Substitutions
Substitution reactions of compounds having an antagonistic orientation of substituents require a more careful analysis. If the directing effects of the two groups oppose each other, the more powerful activating group has the dominant influence, but mixtures of products are often formed. For example, bromination of p -methylphenol yields primarily 2-bromo-4-methylphenol because –OH is a more powerful activator than –CH 3 .
If the substituents are identical, as in example 1 below, the symmetry of the molecule will again simplify the decision. When one substituent has a pair of non-bonding electrons available for adjacent charge stabilization, it will normally exert the product determining influence, examples 2, 4 & 5, even though it may be overall deactivating (case 2). Case 3 reflects a combination of steric hindrance and the superior innate stabilizing ability of methyl groups relative to other alkyl substituents. Example 6 is interesting in that it demonstrates the conversion of an activating ortho/para-directing group into a deactivating meta-directing "onium" cation [–NH(CH 3 ) 2 (+) ] in a strong acid environment.
Exercises
At what position would you expect electrophilic substitution to occur in each of the following substances:
- b. c.
- Answer
-
- Ortho and para to − OCH 3
- Ortho and para to − NH 2
- Ortho and para to − Cl
Predict the products of the following reactions:
- Answer
-
Show the major product(s) from reaction of the following substances with (1) CH 3 CH 2 C(O)Cl, AlCl 3 and (2) SO 3 , H 2 SO 4 :
- Answer
-
The reactions occur ortho and para to the − CH 3 group.


1.E: Trigonometric Functions (Exercises)
1.1: Angles
In this section, we will examine properties of angles.
Verbal
1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.
- Answer
-
2) Explain why there are an infinite number of angles that are coterminal to a certain angle.
3) State what a positive or negative angle signifies, and explain how to draw each.
- Answer
-
Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.
4) How does radian measure of an angle compare to the degree measure? Include an explanation of \(1\) radian in your paragraph.
5) Explain the differences between linear speed and angular speed when describing motion along a circular path.
- Answer
-
Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.
Graphical
For the exercises 6-21, draw an angle in standard position with the given measure.
6) \(30^{\circ}\)
7) \(300^{\circ}\)
- Answer
-
8) \(-80^{\circ}\)
9) \(135^{\circ}\)
- Answer
-
10) \(-150^{\circ}\)
11) \(\dfrac{2π}{3}\)
- Answer
-
12) \(\dfrac{7π}{4}\)
13) \(\dfrac{5π}{6}\)
- Answer
-
14) \(\dfrac{π}{2}\)
15) \(−\dfrac{π}{10}\)
- Answer
-
16) \(415^{\circ}\)
17) \(-120^{\circ}\)
- Answer
-
\(240^{\circ}\)
18) \(-315^{\circ}\)
19)\(\dfrac{22π}{3}\)
- Answer
-
\(\dfrac{4π}{3}\)
20) \(−\dfrac{π}{6}\)
21) \(−\dfrac{4π}{3}\)
- Answer
-
\(\dfrac{2π}{3}\)
For the exercises 22-23, refer to Figure below. Round to two decimal places.
22) Find the arc length.
23) Find the area of the sector.
- Answer
-
\(\dfrac{27π}{2}≈11.00 \text{ in}^2\)
For the exercises 24-25, refer to Figure below. Round to two decimal places.
24) Find the arc length.
25) Find the area of the sector.
- Answer
-
\(\dfrac{81π}{20}≈12.72\text{ cm}^2\)
Algebraic
For the exercises 26-32, convert angles in radians to degrees.
26) \(\dfrac{3π}{4}\) radians
27) \(\dfrac{π}{9}\) radians
- Answer
-
\(20^{\circ}\)
28) \(−\dfrac{5π}{4}\) radians
29) \(\dfrac{π}{3}\) radians
- Answer
-
\(60^{\circ}\)
30) \(−\dfrac{7π}{3}\) radians
31) \(−\dfrac{5π}{12}\) radians
- Answer
-
\(-75^{\circ}\)
32) \(\dfrac{11π}{6}\) radians
For the exercises 33-39, convert angles in degrees to radians.
33) \(90^{\circ}\)
- Answer
-
\(\dfrac{π}{2}\) radians
34) \(100^{\circ}\)
35) \(-540^{\circ}\)
- Answer
-
\(−3π\) radians
36) \(-120^{\circ}\)
37) \(180^{\circ}\)
- Answer
-
\(π\) radians
38) \(-315^{\circ}\)
39) \(150^{\circ}\)
- Answer
-
\(\dfrac{5π}{6}\) radians
For the exercises 40-45, use to given information to find the length of a circular arc. Round to two decimal places.
40) Find the length of the arc of a circle of radius \(12\) inches subtended by a central angle of \(\dfrac{π}{4}\) radians.
41) Find the length of the arc of a circle of radius \(5.02\) miles subtended by the central angle of \(\dfrac{π}{3}\).
- Answer
-
\(\dfrac{5.02π}{3}≈5.26\) miles
42) Find the length of the arc of a circle of diameter \(14\) meters subtended by the central angle of \(\dfrac{5\pi }{6}\).
43) Find the length of the arc of a circle of radius \(10\) centimeters subtended by the central angle of \(50^{\circ}\).
- Answer
-
\(\dfrac{25π}{9}≈8.73\) centimeters
44) Find the length of the arc of a circle of radius \(5\) inches subtended by the central angle of \(220^{circ}\).
45) Find the length of the arc of a circle of diameter \(12\) meters subtended by the central angle is \(63^{circ}\).
- Answer
-
\(\dfrac{21π}{10}≈6.60\) meters
For the exercises 46-49, use the given information to find the area of the sector. Round to four decimal places.
46) A sector of a circle has a central angle of \(45^{\circ}\) and a radius \(6\) cm.
47) A sector of a circle has a central angle of \(30^{\circ}\) and a radius of \(20\) cm.
- Answer
-
\(104.7198\; cm^2\)
48) A sector of a circle with diameter \(10\) feet and an angle of \(\dfrac{π}{2}\) radians.
49) A sector of a circle with radius of \(0.7\) inches and an angle of \(π\) radians.
- Answer
-
\(0.7697\; in^2\)
For the exercises 50-53, find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal to the given angle.
50) \(-40^{\circ}\)
51) \(-110^{\circ}\)
- Answer
-
\(250^{\circ}\)
52) \(700^{\circ}\)
53) \(1400^{\circ}\)
- Answer
-
\(320^{\circ}\)
For the exercises 54-57, find the angle between \(0\) and \(2\pi \) in radians that is coterminal to the given angle.
54) \(−\dfrac{π}{9}\)
55) \(\dfrac{10π}{3}\)
- Answer
-
\(\dfrac{4π}{3}\)
56) \(\dfrac{13π}{6}\)
57) \(\dfrac{44π}{9}\)
- Answer
-
\(\dfrac{8π}{9}\)
Real-World Applications
58) A truck with \(32\)-inch diameter wheels is traveling at \(60\) mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?
59) A bicycle with \(24\)-inch diameter wheels is traveling at \(15\) mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?
- Answer
-
\(1320\) rad \(210.085\) RPM
60) A wheel of radius \(8\) inches is rotating \(15^{\circ}/s\). What is the linear speed \(v\), the angular speed in RPM, and the angular speed in rad/s?
61) A wheel of radius \(14\) inches is rotating \(0.5 \text{rad/s}\). What is the linear speed \(v\), the angular speed in RPM, and the angular speed in deg/s?
- Answer
-
\(7\) in./s, \(4.77\) RPM, \(28.65\) deg/s
62) A CD has diameter of \(120\) millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about \(200\) RPM (revolutions per minute). Find the linear speed.
63) When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about \(4800\) RPM (revolutions per minute). Find the linear speed if the CD has diameter of \(120\) millimeters.
- Answer
-
\(1,809,557.37 \text{ mm/min}=30.16 \text{ m/s}\)
64) A person is standing on the equator of Earth (radius \(3960\) miles). What are his linear and angular speeds?
65) Find the distance along an arc on the surface of Earth that subtends a central angle of \(5\) minutes \((1 \text{ minute}=\dfrac{1}{60} \text{ degree})\). The radius of Earth is \(3960\) miles.
- Answer
-
\(5.76\) miles
66) Find the distance along an arc on the surface of Earth that subtends a central angle of \(7\) minutes \((1 \text{ minute}=\dfrac{1}{60} \text{ degree})\). The radius of Earth is \(3960\) miles.
67) Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in \(20\) minutes?
- Answer
-
\(120°\)
Extensions
68) Two cities have the same longitude. The latitude of city A is \(9.00\) degrees north and the latitude of city B is \(30.00\) degree north. Assume the radius of the earth is \(3960\) miles. Find the distance between the two cities.
69) A city is located at \(40\) degrees north latitude. Assume the radius of the earth is \(3960\) miles and the earth rotates once every \(24\) hours. Find the linear speed of a person who resides in this city.
- Answer
-
\(794\) miles per hour
70) A city is located at \(75\) degrees north latitude. Assume the radius of the earth is \(3960\) miles and the earth rotates once every \(24\) hours. Find the linear speed of a person who resides in this city.
71) Find the linear speed of the moon if the average distance between the earth and moon is \(239,000\) miles, assuming the orbit of the moon is circular and requires about \(28\) days. Express answer in miles per hour.
- Answer
-
\(2,234\) miles per hour
72) A bicycle has wheels \(28\) inches in diameter. A tachometer determines that the wheels are rotating at \(180\) RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.
73) A car travels \(3\) miles. Its tires make \(2640\) revolutions. What is the radius of a tire in inches?
- Answer
-
\(11.5\) inches
74) A wheel on a tractor has a \(24\)-inch diameter. How many revolutions does the wheel make if the tractor travels \(4\) miles?
1.2: Unit Circle - Sine and Cosine Functions
Verbal
1) Describe the unit circle.
- Answer
-
The unit circle is a circle of radius \(1\) centered at the origin.
2) What do the \(x\)- and \(y\)-coordinates of the points on the unit circle represent?
3) Discuss the difference between a coterminal angle and a reference angle.
- Answer
-
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, \(t\), formed by the terminal side of the angle \(t\) and the horizontal axis.
4) Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.
5) Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.
- Answer
-
The sine values are equal.
Algebraic
For the exercises 6-9, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by \(t\) lies.
6) \( \sin (t)<0\) and \( \cos (t)<0\)
7) \( \sin (t)>0\) and \( \cos (t)>0\)
- Answer
-
\(\textrm{I}\)
8) \( \sin (t)>0 \) and \( \cos (t)<0\)
9) \( \sin (t)<0 \) and \( \cos (t)>0\)
- Answer
-
\(\textrm{IV}\)
For the exercises 10-22, find the exact value of each trigonometric function.
10) \(\sin \dfrac{π}{2}\)
11) \(\sin \dfrac{π}{3}\)
- Answer
-
\(\dfrac{\sqrt{3}}{2}\)
12) \( \cos \dfrac{π}{2}\)
13) \( \cos \dfrac{π}{3}\)
- Answer
-
\(\dfrac{1}{2}\)
14) \( \sin \dfrac{π}{4}\)
15) \( \cos \dfrac{π}{4}\)
- Answer
-
\(\dfrac{\sqrt{2}}{2}\)
16) \( \sin \dfrac{π}{6}\)
17) \( \sin π\)
- Answer
-
\(0\)
18) \( \sin \dfrac{3π}{2}\)
19) \( \cos π\)
- Answer
-
\(−1\)
20) \( \cos 0\)
21) \(cos \dfrac{π}{6}\)
- Answer
-
\(\dfrac{\sqrt{3}}{2}\)
22) \( \sin 0\)
Numeric
For the exercises 23-33, state the reference angle for the given angle.
23) \(240°\)
- Answer
-
\(60°\)
24) \(−170°\)
25) \(100°\)
- Answer
-
\(80°\)
26) \(−315°\)
27) \(135°\)
- Answer
-
\(45°\)
28) \(\dfrac{5π}{4}\)
29) \(\dfrac{2π}{3}\)
- Answer
-
\(\dfrac{π}{3}\)
30) \(\dfrac{5π}{6}\)
31) \(−\dfrac{11π}{3}\)
- Answer
-
\(\dfrac{π}{3}\)
32) \(\dfrac{−7π}{4}\)
33) \(\dfrac{−π}{8}\)
- Answer
-
\(\dfrac{π}{8}\)
For the exercises 34-49, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.
34) \(225°\)
35) \(300°\)
- Answer
-
\(60°\), Quadrant IV, \( \sin (300°)=−\dfrac{\sqrt{3}}{2}, \cos (300°)=\dfrac{1}{2}\)
36) \(320°\)
37) \(135°\)
- Answer
-
\(45°\), Quadrant II, \( \sin (135°)=\dfrac{\sqrt{2}}{2}, \cos (135°)=−\dfrac{\sqrt{2}}{2}\)
38) \(210°\)
39) \(120°\)
- Answer
-
\(60°\), Quadrant II, \(\sin (120°)=\dfrac{\sqrt{3}}{2}\), \(\cos (120°)=−\dfrac{1}{2}\)
40) \(250°\)
41) \(150°\)
- Answer
-
\(30°\), Quadrant II, \( \sin (150°)=\frac{1}{2}\), \(\cos(150°)=−\dfrac{\sqrt{3}}{2}\)
42) \(\dfrac{5π}{4}\)
43) \(\dfrac{7π}{6}\)
- Answer
-
\(\dfrac{π}{6}\), Quadrant III, \(\sin \left( \dfrac{7π}{6}\right )=−\dfrac{1}{2}\), \(\cos \left (\dfrac{7π}{6} \right)=−\dfrac{\sqrt{3}}{2}\)
44) \(\dfrac{5π}{3}\)
45) \(\dfrac{3π}{4}\)
- Answer
-
\(\dfrac{π}{4}\), Quadrant II, \(\sin \left(\dfrac{3π}{4}\right)=\dfrac{\sqrt{2}}{2}\), \(\cos\left(\dfrac{4π}{3}\right)=−\dfrac{\sqrt{2}}{2}\)
46) \(\dfrac{4π}{3}\)
47) \(\dfrac{2π}{3}\)
- Answer
-
\(\dfrac{π}{3}\), Quadrant II, \( \sin \left(\dfrac{2π}{3}\right)=\dfrac{\sqrt{3}}{2}\), \( \cos \left(\dfrac{2π}{3}\right)=−\dfrac{1}{2}\)
48) \(\dfrac{5π}{6}\)
49) \(\dfrac{7π}{4}\)
- Answer
-
\(\dfrac{π}{4}\), Quadrant IV, \( \sin \left(\dfrac{7π}{4}\right)=−\dfrac{\sqrt{2}}{2}\), \( \cos \left(\dfrac{7π}{4}\right)=\dfrac{\sqrt{2}}{2}\)
For the exercises 50-59, find the requested value.
50) If \(\cos (t)=\dfrac{1}{7}\) and \(t\) is in the \(4^{th}\) quadrant, find \( \sin (t)\).
51) If \( \cos (t)=\dfrac{2}{9}\) and \(t\) is in the \(1^{st}\) quadrant, find \(\sin (t)\).
- Answer
-
\(\dfrac{\sqrt{77}}{9}\)
52) If \(\sin (t)=\dfrac{3}{8}\) and \(t\) is in the \(2^{nd}\) quadrant, find \( \cos (t)\).
53) If \( \sin (t)=−\dfrac{1}{4}\) and \(t\) is in the \(3^{rd}\) quadrant, find \(\cos (t)\).
- Answer
-
\(−\dfrac{\sqrt{15}}{4}\)
54) Find the coordinates of the point on a circle with radius \(15\) corresponding to an angle of \(220°\).
55) Find the coordinates of the point on a circle with radius \(20\) corresponding to an angle of \(120°\).
- Answer
-
\((−10,10\sqrt{3})\)
56) Find the coordinates of the point on a circle with radius \(8\) corresponding to an angle of \(\dfrac{7π}{4}\).
57) Find the coordinates of the point on a circle with radius \(16\) corresponding to an angle of \(\dfrac{5π}{9}\).
- Answer
-
\((–2.778,15.757)\)
58) State the domain of the sine and cosine functions.
59) State the range of the sine and cosine functions.
- Answer
-
\([–1,1]\)
Graphical
For the exercises 60-79, use the given point on the unit circle to find the value of the sine and cosine of \(t\).
60)
61)
- Answer
-
\( \sin t=\dfrac{1}{2}, \cos t=−\dfrac{\sqrt{3}}{2}\)
62)
63)
- Answer
-
\( \sin t=− \dfrac{\sqrt{2}}{2}, \cos t=−\dfrac{\sqrt{2}}{2}\)
64)
65)
- Answer
-
\( \sin t=\dfrac{\sqrt{3}}{2},\cos t=−\dfrac{1}{2}\)
66)
67)
- Answer
-
\( \sin t=− \dfrac{\sqrt{2}}{2}, \cos t=\dfrac{\sqrt{2}}{2}\)
68)
69)
- Answer
-
\( \sin t=0, \cos t=−1\)
70)
71)
- Answer
-
\( \sin t=−0.596, \cos t=0.803\)
72)
73)
- Answer
-
\(\sin t=\dfrac{1}{2}, \cos t= \dfrac{\sqrt{3}}{2}\)
74)
75)
- Answer
-
\( \sin t=−\dfrac{1}{2}, \cos t= \dfrac{\sqrt{3}}{2} \)
76)
77)
- Answer
-
\( \sin t=0.761, \cos t=−0.649 \)
78)
79)
- Answer
-
\( \sin t=1, \cos t=0\)
Technology
For the exercises 80-89, use a graphing calculator to evaluate.
80) \( \sin \dfrac{5π}{9}\)
81) \(cos \dfrac{5π}{9}\)
- Answer
-
\(−0.1736\)
82) \( \sin \dfrac{π}{10}\)
83) \( \cos \dfrac{π}{10}\)
- Answer
-
\(0.9511\)
84) \( \sin \dfrac{3π}{4}\)
85) \(\cos \dfrac{3π}{4}\)
- Answer
-
\(−0.7071\)
86) \( \sin 98° \)
87) \( \cos 98° \)
- Answer
-
\(−0.1392\)
88) \( \cos 310° \)
89) \( \sin 310° \)
- Answer
-
\(−0.7660\)
Extensions
For the exercises 90-99, evaluate.
90) \( \sin \left(\dfrac{11π}{3}\right) \cos \left(\dfrac{−5π}{6}\right)\)
91) \( \sin \left(\dfrac{3π}{4}\right) \cos \left(\dfrac{5π}{3}\right) \)
- Answer
-
\(\dfrac{\sqrt{2}}{4}\)
92) \( \sin \left(− \dfrac{4π}{3}\right) \cos \left(\dfrac{π}{2}\right)\)
93) \( \sin \left(\dfrac{−9π}{4}\right) \cos \left(\dfrac{−π}{6}\right)\)
- Answer
-
\(−\dfrac{\sqrt{6}}{4}\)
94) \( \sin \left(\dfrac{π}{6}\right) \cos \left(\dfrac{−π}{3}\right) \)
95) \( \sin \left(\dfrac{7π}{4}\right) \cos \left(\dfrac{−2π}{3}\right) \)
- Answer
-
\(\dfrac{\sqrt{2}}{4}\)
96) \( \cos \left(\dfrac{5π}{6}\right) \cos \left(\dfrac{2π}{3}\right)\)
97) \( \cos \left(\dfrac{−π}{3}\right) \cos \left(\dfrac{π}{4}\right) \)
- Answer
-
\(\dfrac{\sqrt{2}}{4}\)
98) \( \sin \left(\dfrac{−5π}{4}\right) \sin \left(\dfrac{11π}{6}\right)\)
99) \( \sin (π) \sin \left(\dfrac{π}{6}\right) \)
- Answer
-
\(0\)
Real-World Applications
For the exercises 100-104, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point \((0,1)\), that is, on the due north position. Assume the carousel revolves counter clockwise.
100) What are the coordinates of the child after \(45\) seconds?
101) What are the coordinates of the child after \(90\) seconds?
- Answer
-
\((0,–1)\)
102) What is the coordinates of the child after \(125\) seconds?
103) When will the child have coordinates \((0.707,–0.707)\) if the ride lasts \(6\) minutes? (There are multiple answers.)
- Answer
-
\(37.5\) seconds, \(97.5\) seconds, \(157.5\) seconds, \(217.5\) seconds, \(277.5\) seconds, \(337.5\) seconds
104) When will the child have coordinates \((−0.866,−0.5)\) if the ride last \(6\) minutes?
1.3: The Other Trigonometric Functions
Verbal
1) On an interval of \([ 0,2π )\), can the sine and cosine values of a radian measure ever be equal? If so, where?
- Answer
-
Yes, when the reference angle is \(\dfrac{π}{4}\) and the terminal side of the angle is in quadrants I and III. Thus, at \(x=\dfrac{π}{4},\dfrac{5π}{4}\), the sine and cosine values are equal.
2) What would you estimate the cosine of \(\pi \) degrees to be? Explain your reasoning.
3) For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
- Answer
-
Substitute the sine of the angle in for \(y\) in the Pythagorean Theorem \(x^2+y^2=1\). Solve for \(x\) and take the negative solution.
4) Describe the secant function.
5) Tangent and cotangent have a period of \(π\). What does this tell us about the output of these functions?
- Answer
-
The outputs of tangent and cotangent will repeat every \(π\) units.
Algebraic
For the exercises 6-17, find the exact value of each expression.
6) \( \tan \dfrac{π}{6}\)
7) \(\sec \dfrac{π}{6}\)
- Answer
-
\(\dfrac{2\sqrt{3}}{3}\)
8) \( \csc \dfrac{π}{6}\)
9) \( \cot \dfrac{π}{6}\)
- Answer
-
\(\sqrt{3}\)
10) \( \tan \dfrac{π}{4}\)
11) \( \sec \dfrac{π}{4}\)
- Answer
-
\(\sqrt{2}\)
12) \( \csc \dfrac{π}{4}\)
13) \( \cot \dfrac{π}{4}\)
- Answer
-
\(1\)
14) \( \tan \dfrac{π}{3}\)
15) \( \sec \dfrac{π}{3}\)
- Answer
-
\(2\)
16) \( \csc \dfrac{π}{3}\)
17) \( \cot \dfrac{π}{3}\)
- Answer
-
\(\dfrac{\sqrt{3}}{3}\)
For the exercises 18-48, use reference angles to evaluate the expression.
18) \( \tan \dfrac{5π}{6}\)
19) \( \sec \dfrac{7π}{6}\)
- Answer
-
\(−\dfrac{2\sqrt{3}}{3}\)
20) \( \csc \dfrac{11π}{6}\)
21) \( \cot \dfrac{13π}{6}\)
- Answer
-
\(\sqrt{3}\)
22) \( \tan \dfrac{7π}{4}\)
23) \( \sec \dfrac{3π}{4}\)
- Answer
-
\(−\sqrt{2}\)
24) \( \csc \dfrac{5π}{4}\)
25) \( \cot \dfrac{11π}{4}\)
- Answer
-
\(−1\)
26) \( \tan \dfrac{8π}{3}\)
27) \( \sec \dfrac{4π}{3}\)
- Answer
-
\(−2\)
28) \( \csc \dfrac{2π}{3}\)
29) \( \cot \dfrac{5π}{3}\)
- Answer
-
\(−\dfrac{\sqrt{3}}{3}\)
30) \( \tan 225°\)
31) \( \sec 300°\)
- Answer
-
\(2\)
32) \( \csc 150°\)
33) \( \cot 240°\)
- Answer
-
\(\dfrac{\sqrt{3}}{3}\)
34) \( \tan 330°\)
35) \( \sec 120°\)
- Answer
-
\(−2\)
36) \( \csc 210°\)
37) \( \cot 315°\)
- Answer
-
\(−1\)
38) If \( \sin t= \dfrac{3}{4}\), and \(t\) is in quadrant II, find \( \cos t, \sec t, \csc t, \tan t, \cot t \).
39) If \( \cos t=−\dfrac{1}{3},\) and \(t\) is in quadrant III, find \( \sin t, \sec t, \csc t, \tan t, \cot t\).
- Answer
-
If \(\sin t=−\dfrac{2\sqrt{2}}{3}, \sec t=−3, \csc t=−\csc t=−\dfrac{3\sqrt{2}}{4},\tan t=2\sqrt{2}, \cot t= \dfrac{\sqrt{2}}{4}\)
40) If \(\tan t=\dfrac{12}{5},\) and \(0≤t< \dfrac{π}{2}\), find \( \sin t, \cos t, \sec t, \csc t,\) and \(\cot t\).
41) If \( \sin t= \dfrac{\sqrt{3}}{2}\) and \( \cos t=\dfrac{1}{2},\) find \( \sec t, \csc t, \tan t,\) and \( \cot t\).
- Answer
-
\( \sec t=2, \csc t=\csc t=\dfrac{2\sqrt{3}}{3}, \tan t= \sqrt{3}, \cot t= \dfrac{\sqrt{3}}{3}\)
42) If \( \sin 40°≈0.643 \; \cos 40°≈0.766 \; \sec 40°,\csc 40°,\tan 40°, \text{ and } \cot 40°\).
43) If \( \sin t= \dfrac{\sqrt{2}}{2},\) what is the \( \sin (−t)\)?
- Answer
-
\(−\dfrac{\sqrt{2}}{2}\)
44) If \( \cos t= \dfrac{1}{2},\) what is the \( \cos (−t)\)?
45) If \( \sec t=3.1,\) what is the \( \sec (−t)\)?
- Answer
-
\(3.1\)
46) If \( \csc t=0.34,\) what is the \( \csc (−t)\)?
47) If \( \tan t=−1.4,\) what is the \( \tan (−t)\)?
- Answer
-
\(1.4\)
48) If \( \cot t=9.23,\) what is the \( \cot (−t)\)?
Graphical
For the exercises 49-51, use the angle in the unit circle to find the value of the each of the six trigonometric functions.
49)
- Answer
-
\( \sin t= \dfrac{\sqrt{2}}{2}, \cos t= \dfrac{\sqrt{2}}{2}, \tan t=1,\cot t=1,\sec t= \sqrt{2}, \csc t= \csc t= \sqrt{2} \)
50)
51)
- Answer
-
\( \sin t=−\dfrac{\sqrt{3}}{2}, \cos t=−\dfrac{1}{2}, \tan t=\sqrt{3}, \cot t= \dfrac{\sqrt{3}}{3}, \sec t=−2, \csc t=−\csc t=−\dfrac{2\sqrt{3}}{3} \)
Technology
For the exercises 52-61, use a graphing calculator to evaluate.
52) \( \csc \dfrac{5π}{9}\)
53) \( \cot \dfrac{4π}{7}\)
- Answer
-
\(–0.228\)
54) \( \sec \dfrac{π}{10}\)
55) \( \tan \dfrac{5π}{8}\)
- Answer
-
\(–2.414\)
56) \( \sec \dfrac{3π}{4}\)
57) \( \csc \dfrac{π}{4}\)
- Answer
-
\(1.414\)
58) \( \tan 98°\)
59) \( \cot 33°\)
- Answer
-
\(1.540\)
60) \( \cot 140°\)
61) \( \sec 310° \)
- Answer
-
\(1.556\)
Extensions
For the exercises 62-69, use identities to evaluate the expression.
62) If \(\tan (t)≈2.7,\) and \( \sin (t)≈0.94,\) find \( \cos (t)\).
63) If \( \tan (t)≈1.3,\) and \( \cos (t)≈0.61\), find \( \sin (t)\).
- Answer
-
\( \sin (t)≈0.79 \)
64) If \( \csc (t)≈3.2,\) and \( \csc (t)≈3.2,\) and \( \cos (t)≈0.95,\) find \( \tan (t)\).
65) If \( \cot (t)≈0.58,\) and \( \cos (t)≈0.5,\) find \( \csc (t)\).
- Answer
-
\( \csc (t)≈1.16\)
66) Determine whether the function \(f(x)=2 \sin x \cos x\) is even, odd, or neither.
67) Determine whether the function \(f(x)=3 \sin ^2 x \cos x + \sec x\) is even, odd, or neither.
- Answer
-
even
68) Determine whether the function \(f(x)= \sin x −2 \cos ^2 x \) is even, odd, or neither.
69) Determine whether the function \(f(x)= \csc ^2 x+ \sec x\) is even, odd, or neither.
- Answer
-
even
For the exercises 70-71, use identities to simplify the expression.
70) \( \csc t \tan t\)
71) \( \dfrac{\sec t}{ \csc t}\)
- Answer
-
\( \dfrac{ \sin t}{ \cos t}= \tan t\)
Real-World Applications
72) The amount of sunlight in a certain city can be modeled by the function \(h=15 \cos \left(\dfrac{1}{600}d\right),\) where \(h\) represents the hours of sunlight, and \(d\) is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the \(42^{nd}\) day of the year. State the period of the function.
73) The amount of sunlight in a certain city can be modeled by the function \(h=16 \cos \left(\dfrac{1}{500}d\right)\), where \(h\) represents the hours of sunlight, and \(d\) is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the \(267^{th}\) day of the year. State the period of the function.
- Answer
-
\(13.77\) hours, period: \(1000π\)
74) The equation \(P=20 \sin (2πt)+100\) models the blood pressure, \(P\), where \(t\) represents time in seconds.
- Find the blood pressure after \(15\) seconds.
- What are the maximum and minimum blood pressures?
75) The height of a piston, \(h\), in inches, can be modeled by the equation \(y=2 \cos x+6,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55°\).
- Answer
-
\(7.73\) inches
76) The height of a piston, \(h\),in inches, can be modeled by the equation \(y=2 \cos x+5,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55°\).
1.4: Right Triangle Trigonometry
Verbal
1) For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.
- Answer
-
2) When a right triangle with a hypotenuse of \(1\) is placed in the unit circle, which sides of the triangle correspond to the \(x\)- and \(y\)-coordinates?
3) The tangent of an angle compares which sides of the right triangle?
- Answer
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The tangent of an angle is the ratio of the opposite side to the adjacent side.
4) What is the relationship between the two acute angles in a right triangle?
5) Explain the cofunction identity.
- Answer
-
For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.
Algebraic
For the exercises 6-9, use cofunctions of complementary angles.
6) \( \cos (34°)= \sin (\_\_°)\)
7) \( \cos (\dfrac{π}{3})= \sin (\_\_\_) \)
- Answer
-
\(\dfrac{π}{6}\)
8) \( \csc (21°) = \sec (\_\_\_°)\)
9) \( \tan (\dfrac{π}{4})= \cot (\_\_)\)
- Answer
-
\(\dfrac{π}{4}\)
For the exercises 10-16, find the lengths of the missing sides if side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is the hypotenuse.
10) \( \cos B= \dfrac{4}{5},a=10\)
11) \( \sin B= \dfrac{1}{2}, a=20\)
- Answer
-
\(b= \dfrac{20\sqrt{3}}{3},c= \dfrac{40\sqrt{3}}{3}\)
12) \( \tan A= \dfrac{5}{12},b=6\)
13) \( \tan A=100,b=100\)
- Answer
-
\(a=10,000,c=10,000.5\)
14) \(\sin B=\dfrac{1}{\sqrt{3}}, a=2 \)
15) \(a=5, ∡ A=60^∘\)
- Answer
-
\(b=\dfrac{5\sqrt{3}}{3},c=\dfrac{10\sqrt{3}}{3}\)
16) \(c=12, ∡ A=45^∘\)
Graphical
For the exercises 17-22, use Figure below to evaluate each trigonometric function of angle \(A\).
17) \(\sin A\)
- Answer
-
\(\dfrac{5\sqrt{29}}{29}\)
18) \( \cos A \)
19) \( \tan A \)
- Answer
-
\(\dfrac{5}{2}\)
20) \(\csc A \)
21) \( \sec A \)
- Answer
-
\(\dfrac{\sqrt{29}}{2}\)
22) \( \cot A \)
For the exercises 23-,28 use Figure below to evaluate each trigonometric function of angle \(A\).
23) \( \sin A\)
- Answer
-
\(\dfrac{5\sqrt{41}}{41}\)
24) \( \cos A\)
25) \( \tan A \)
- Answer
-
\(\dfrac{5}{4}\)
26) \( \csc A\)
27) \( \sec A\)
- Answer
-
\(\dfrac{\sqrt{41}}{4}\)
28) \(\cot A\)
For the exercises 29-31, solve for the unknown sides of the given triangle.
29)
- Answer
-
\(c=14, b=7\sqrt{3}\)
30)
31)
- Answer
-
\(a=15, b=15 \)
Technology
For the exercises 32-41, use a calculator to find the length of each side to four decimal places.
32)
33)
- Answer
-
\(b=9.9970, c=12.2041\)
34)
35)
- Answer
-
\(a=2.0838, b=11.8177\)
36)
37) \(b=15, ∡B=15^∘\)
- Answer
-
\(a=55.9808,c=57.9555\)
38) \(c=200, ∡B=5^∘\)
39) \(c=50, ∡B=21^∘\)
- Answer
-
\(a=46.6790,b=17.9184\)
40) \(a=30, ∡A=27^∘\)
41) \(b=3.5, ∡A=78^∘\)
- Answer
-
\(a=16.4662,c=16.8341\)
Extensions
42) Find \(x\).
43) Find \(x\).
- Answer
-
\(188.3159\)
44) Find \(x\).
45) Find \(x\).
- Answer
-
\(200.6737\)
46) A radio tower is located \(400\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(36°\), and that the angle of depression to the bottom of the tower is \(23°\). How tall is the tower?
47) A radio tower is located \(325\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(43°\), and that the angle of depression to the bottom of the tower is \(31°\). How tall is the tower?
- Answer
-
\(498.3471\) ft
48) A \(200\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(15°\), and that the angle of depression to the bottom of the tower is \(2°\). How far is the person from the monument?
49) A \(400\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(18°\), and that the angle of depression to the bottom of the monument is \(3°\). How far is the person from the monument?
- Answer
-
\(1060.09\) ft
50) There is an antenna on the top of a building. From a location \(300\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(40°\). From the same location, the angle of elevation to the top of the antenna is measured to be \(43°\). Find the height of the antenna.
51) There is lightning rod on the top of a building. From a location \(500\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(36°\). From the same location, the angle of elevation to the top of the lightning rod is measured to be \(38°\). Find the height of the lightning rod.
- Answer
-
\(27.372\) ft
Real-World Applications
52) A \(33\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?
53) A \(23\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?
- Answer
-
\(22.6506\) ft
54) The angle of elevation to the top of a building in New York is found to be \(9\) degrees from the ground at a distance of \(1\) mile from the base of the building. Using this information, find the height of the building.
55) The angle of elevation to the top of a building in Seattle is found to be \(2\) degrees from the ground at a distance of \(2\) miles from the base of the building. Using this information, find the height of the building.
- Answer
-
\(368.7633\) ft
56) Assuming that a \(370\)-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be \(60°\), how far from the base of the tree am I?
Contributor
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


4.10: Pedagogy
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The term pedagogy refers to the holistic nature of educators’ professional practice (especially those aspects that involve building and nurturing relationships), decision-making, and teaching and learning. When educators establish respectful relationships with children and families, they are able to work together to develop programs and experiences which are relevant to children and build on individual and group interests. These experiences create possibilities for children’s own ideas and activities, allowing them to celebrate their own interests and friendships and express themselves in different ways.
Educators’ professional judgments are central to their active role in facilitating children’s learning. In making professional judgements, they weave together their:
They also draw on their creativity, imagination and insight to help them improvise and adjust their practice to suit the time, place and context of learning.
Different theories about childhood inform approaches to children’s learning and development. School-age care educators draw upon a range of perspectives in their work. Drawing on a range of perspectives and theories can challenge traditional ways of seeing children, facilitating learning, and encourage educators, as individuals and with colleagues, to: