2.3: Chapter 6: STEM Beyond the Acronym: Ethical Considerations in Standardizing STEM Education in K-12
The American education system has undergone numerous iterations over the years. From the No Child Left Behind Act of 2002 to the Back to Basics movement (and it’s emphasis on the 3 R’s — reading, ‘riting, ‘rithmatic), educational trends have influenced decision-makers to design and build programs in the hopes of improving student outcomes on standardized tests. Over the past few decades, STEM (Science, Technology, Engineering, and Math) has been a widely discussed program in education (Blackley & Howell, 2015; Breiner et al., 2012; Bybee, 2013). However, STEM has yet to be truly defined, since government officials, teachers, parents, and other stakeholders have varying perspectives of what STEM is and how it should look in the classroom (Breiner et al., 2012). STEM is viewed in a variety of ways that are dependent on stakeholders and on ethical considerations — which causes tension when trying to create a unilateral definition in order to better direct teaching practice.
Some argue that standardizing STEM would result in teaching silos (Blackley & Howell, 2015; Bybee, 2013) with little diversity of thinking, leading to STEM subjects being taught as four separate subjects with assessments conducted through standardized testing. Blackely and Howell (2015) argue that “teachers have defaulted to the notion of S.T.E.M. rather than STEM,” where the full stops “signif[y] . . . the silo-ing of the four distinct discipline areas, rather than their integration” (p. 104). By teaching science, technology, engineering, and math as separate subjects (or silos), the learning experience shifts from being interdisciplinary, transdisciplinary, and integrated, to one that is focused on an individualized curriculum (Bybee, 2013) that “reflects decisions made centuries ago, especially around science and mathematics” (Davis et al., 2019, p. 3). Others argue that to truly meet the needs of learners, a foundational approach or mindset that weaves STEM throughout all subject areas is crucial. This orientation creates a comprehensive curriculum that utilizes a transdisciplinary approach with an objective that “resolve[s] real world or complex problems . . . provide[s] different perspectives on problems” (Choi & Pak, 2006, p. 351), and requires insights and perspectives from more than one discipline (Davis et al., 2019).
This chapter will consider the ethical implications of STEM education (silo versus mindset approach) in the US and its impact on learners, with special consideration for gender, ethnicity, and socioeconomic standing. When determining how to define STEM education, who should be responsible for determining best practices and the future direction that education should take? Decisions need to be made to meet the needs of the learners themselves, not based on funding nor global positioning. There are multiple ethical implications when creating a standardized curriculum for STEM without considering all of the stakeholders (i.e. students, parents, and educators). This standardization could create a larger achievement gap among minority groups, girls, and those in lower socioeconomic districts. The push for a standardized definition of STEM as a silo, with standardized testing being the gauge of effectiveness, has illustrated the disparity between gender, socioeconomic, and ethnic groups. In the US, various programs have been designed to promote STEM education for girls [New Tab] and minority groups [New Tab] ; however, males and the privileged classes still continue to dominate in STEM careers and industries (Building America’s Future, n.d.; Chang, 2019; Maclean, 2017).
Full Disclosure
STEM is an umbrella concept that incorporates Science, Technology, Engineering, and Math. How STEM education is actually incorporated into different schools and districts varies. The original implementation of SMET in the US was led by the National Science Foundation (NSF); the acronym was later changed to STEM (Bybee, 2013; Sanders, 2009) in the 1990s in response to a need to strengthen science and math education. Until the early 2000s, few people understood the term STEM and many linked it to stem cell research and plants (Bybee, 2013; Sanders, 2009). STEM, when taken verbatim, refers only to the four subject areas. For the purpose of this chapter, the focus will be on STEM education, which is an important distinction.
One impetus for the growing STEM education trend came in reaction to the results of the early PISA [New Tab] test results. The scores on these tests, in addition to a belief in a shortage of people to work in STEM-related fields, prompted the implementation of STEM education in K-12 schools around the world (Williams, 2011). In the US, STEM education became a focused trend to shift “American students from the middle to the top of the pack [internationally] in science and math over the next decade” (The White House, 2009, para. 1) by raising test scores. Another priority was to meet the perceived STEM crisis — a lack of skilled workforce to fill the growing specialized job market (Jadav, 2013; The STEM Crisis & Our Solution, 2020; Xu, 2015). The Partnership for 21st Century Skills (2008) claimed that STEM education was the key to success in the 21st Century, and that without a “charge to action,” the US would be left behind in the global market. This charge to action has led many schools in the US to focus primarily on teaching each subject as a silo, with small pockets of schools and districts forging their own mindset path (Cherry Creek School District [CCSD], n.d.). This push for STEM education raises many ethical concerns regarding privacy, autonomy, and best practices for our students.
STEM as a foundational way of thinking focuses on students iterating; collaborating; and using science, technology, engineering, math, and humanities in an interdisciplinary way to solve real-world problems (Davis, 2019). STEM as an inquiry-based integrated method of teaching these subjects is not necessarily a new concept. In the 1990s, the American Association for the Advancement of Science (2009) published a series of benchmarks for science literacy. They believed that:
The common core or learning in science, math, and technology
should focus on science literacy, not on an understanding of each
of the separate disciplines. Moreover, the core studies should include
connections among science, math, and technology and between
those areas and the arts and humanities and the vocational subjects. (p. xii)
Sanders (2009) argues that STEM should be taught with the mindset of “purposeful design and inquiry” (p.21), which incorporates all areas within the STEM umbrella. The integration of the disciplines can provide students with the opportunity to explore and develop potential solutions to global issues, such as renewable energy sources and climate change (Davis, 2019). In districts such as Cherry Creek Schools in Centennial, Colorado, this remains the focus of their STEM program. They see STEM as a focused way of teaching that “instills a deep and extensive understanding of STEM content applied in real-world contexts” (Cherry Creek Schools District, n.d., para. 2).
The term STEM has been used and reused so many times that its meaning has become ambiguous (Bybee, 2013; Sanders, 2009). One school’s STEM program can look very different from another. One school can have access to robotics, 3D printers, and one-to-one student laptops. Their program can focus on the T of STEM but have little to no incorporation of design thinking or science and math (indicating a silo approach). Another school can have very little technology but has students ideating, building, and conversing about real-world problems (foundational thinking). As there is no universal definition of STEM, there is a wide variety of ways in which it is being incorporated into schools. From this ambiguity emerge several ethical issues that need to be considered.
Connection of STEM Education in Teaching and Learning to Privacy, Data Security, and Informed Consent
When determining a standardized definition for STEM, the agenda of stakeholders can greatly influence the direction this standardization takes. One way of influencing how STEM is defined is through funding. Funding from sources such as Google, Code.org, and other companies in Silicon Valley has been influential in pushing STEM education in the US (Caperton, 2012), as illustrated by the most recent $300 million of funding in 2017. The motives behind a companies’ investment in STEM education can be considered from an ethical perspective. Some argue that Silicon Valley and subsequent tech giants are pushing their own agendas and have ulterior motives for this funding. One such agenda focuses on the money that can be made by pushing the T in STEM education, which in many iterations refers only to computing. When schools focus on the T in STEM, they essentially buy into the programs and products on the market. This push for technology can lessen the time spent in the collective work needed to unify STEM as a foundational way of thinking, and can create a market in which money must be spent in order to ensure that schools have the latest and greatest technology — technology that continually changes and requires constant upgrading.
Through its ease of use, more affordable devices, and an easily manageable platform, Google for Education [New Tab] , which includes Google Classroom and G Suite, has been adopted by many school districts around the US, as well as other countries around the world. It has been marketed as a platform that “focuses on creating the best educational experience for over 70 million students and teachers in more than 180 countries” ( Google for Education, n.d.) . However, some critics consider whether this accessibility is really free. Tech giants, such as Google, have digital dominance and can effect the economy, society, and politics (Moore & Tambini, 2018), and education (Villapaz, 2014). For example, teachers can use Google to manage classes, share homework, and create shareable, collaborative projects for students without any cost to the school or parents. Ethically, it is important to consider the possible benefits Google might receive with the “free” access provided to students and teachers. Google has faced many lawsuits over the years due to its data mining without parental consent. In September 2019, Google paid out $170 million to the state of New York [New Tab] in response to allegations that Google and YouTube “earned millions by illegally collecting personal information from children without their parents’ consent” (Elias & Feiner, 2019, para. 1). More recently, new lawsuits have been filed by New Mexico’s state attorney general [New Tab] alleging that “the tech company is illegally collecting personal data generated by children in violation of federal and state laws” (Bryan, 2020).
Ethical Considerations of Privacy, Data, and Security When Schools and Districts Focus on the T (Computing) in STEM
When considering the emphasis of the T in STEM education from a deontological ethical perspective (Farrow, 2016) or in term of moral obligation, the T could be considered a money-making strategy employed by big business, most notably big business out of Silicon Valley. Through the use of its devices and free apps, Google has been collecting information from and about students under the age of 13. Their data mining includes [New Tab] “search history and which results students click on, videos they search for and watch on YouTube, usage data and preferences, Gmail messages, G+ profiles and photos, docs, and other Google-hosted content, and content that flows through Google’s systems” (Electronic Frontier Foundation, n.d., para. 7). The means through which Google is learning about future consumers are justifiable, as they are a money-making enterprise. They are using this information to build profiles and to focus advertisements on non-education suite websites such as YouTube. Through these profiles, Google can expand its database and focus advertisements to target consumerism; this, in turn, brings in revenue for the company. This mining of children’s data could be considered a justifiable means to an end. The revenue that companies like Google earn far outweighs the ethical implications of collecting this data.
From a consequentialist ethical perspective (Farrow, 2016), the consequences — the invasion of student privacy with and without parental consent by these companies — far outweigh the financial gain. There are state and federal acts, such as the Family Educational Rights and Privacy Act of 1974 (FERPA) and the Children’s Online Privacy Protection Act of 1998 (COPPA), in place that work to protect minors from the collection and use of their private information. However, these policies are limited in their scope and can at times be ineffective at protecting students (Regan & Jesse, 2019). The lack of transparency from big data companies regarding their collection and use of the data makes it extremely difficult for lawmakers to prove probable cause (Regan & Jesse, 2019). The nature of technology and the tightly guarded algorithms that companies use have created a new world, where laws and acts to protect children are ultimately ineffective. The continually changing nature of technology and the sophistication of big data algorithms make it difficult for parents and minors to truly know what they are agreeing to when they click “agree” below the terms of use. The legalese and sophisticated language mask the underlying agreement. When students and parents agree, do they truly understand what they are agreeing to? Not only are minors being tracked and profiled within Google’s education suite, they are also losing their autonomy. With access to photos, emails, Google maps, etc. — student anonymity is being lost.
Is it ethical for companies to use information garnered from minors to earn profit and nudge them towards particular brands or advertisements? From a virtues perspective (Farrow, 2016), it is an individual’s right to full disclosure, privacy, and autonomy, especially when considering those under the age of majority. Parents, schools, and government bodies are morally obligated to protect those who cannot protect themselves, and to act ethically. Children essentially lose this protection when the door is opened for big data companies to access a their data, parents, schools, and government bodies. Educational use of software and hardware, such as ChromeBooks and Google for Education, invites these companies into our schools under the guise of supporting learners. However, there are potential conflicts of interest as edtech companies work with the schools and teachers while at the same time working with districts, researchers, and shareholders. Is this conflict of interest working to protect our students?
Implications
When STEM is defined with the T as its focus, students become vulnerable to considerable issues around privacy, consent, and autonomy. Through the use of programs such as Google Education, students and teachers receive easy access to a wealth of free online apps advertised to make online learning simpler and more collaborative. These apps grant students and teachers the ability to access a plethora of online programs for free. However, through signing up and clicking “I accept” to the terms of use for each of these programs, students unwittingly give access to companies to data-mine and track every click. When we focus on the T, money is continually funnelled back to companies in Silicon Valley and to other tech giants through the purchasing of licenses, laptops, iPads, and robots. In addition to school spending, these companies profit from advertisements are driven by clickbait and tracked student searches. If STEM is defined as a mindset, however, the use of and reliance on technology becomes lessened, as students look at real-world problems from an interdisciplinary lens, not filtered through technology.
Connection of STEM Education in Teaching to Educational Integrity by Avoiding Harm and Minimizing Risk
Since the early 2000s, government funding programs aimed at improving test scores and deepening the teaching of STEM in K-12 and postsecondary institutions (President’s Council of Advisors on Science and Technology, 2010). To meet these needs, the federal government has viewed STEM not as a mindset, but rather as the strengthening of each individual subject under its own umbrella (as a silo). Defining STEM as separate subjects allows for a greater focus on the core subjects and creates a standardization of practice. This standardization can lead to a standardized assessment tool (tests) that have been used historically to track student progress and could lead to gradually increasing sanctions and/or school closures (Duignan & Nolen, 2019; Famularo et al., 2013).
This standardization of practice does not always consider gender, ethnicity, or socioeconomic status. When responding to multiple-choice test questions, students are unable to bring their social narratives or real-life experiences to bear. Rather, only one answer is correct, thus limiting diversity of thinking and providing no option for students to think outside of the box to solve the problem. Examining these initiatives reveals that they have not changed teaching of STEM subjects much at the classroom level, nor have they greatly increased student interest in the subjects (Breiner et al., 2012). When considering these initiatives and the limited improvement of test scores on a global scale, there appears to be a disconnect between these policies and the reality of teaching 21st-century students. This raises the question of who should be making these policy decisions.
Ethical Considerations When Determining a Standardized Definition of STEM as a Silo
From a deontological ethical perspective (Farrow, 2016), this standardization of practice and assessment will allow the US to track students and educators which will then help determine who is underperforming and what sanctions are needed to improve individual schools and districts. The intention of these sanctions (Duignan & Nolen, 2019) is to push educators to better prepare their students in the core subjects (science, math, and reading). This view argues that the means to which test scores are improved outweighs the moral implications.
On the other hand, if we look at the standardization of STEM into its component subjects from a consequentialist ethical viewpoint (Farrow, 2016), data from the 2018 Programme for International Student Assessment (OECD) shows that there has been no major improvement in science and math test scores in the US even with massive federal funding over the past twenty years for STEM education. From this perspective, pushing STEM education as separate core subjects has not resulted in improved of test scores, and the US continues to rest solidly in the middle of the pack.
According to ethical virtue theory (Farrow, 2016) the standardization of STEM education is detrimental to our students’ learning. By narrowing the definition of STEM to its component subjects, students are not encouraged to tackle real-world problems from an interdisciplinary standpoint. Students are relegated to a number on a paper that does not take into consideration diversity, culture, nor gender. Should one test written each year determine the future educational trajectory of a student? From this perspective, students should be assessed using a variety of tools and instruments that can effectively account for multiculturalism, ability, and diversity of thinking. A standardization of STEM and subsequent assessment has limited ability to bring in multiculturalism and divergent thinking. This brings to light a need for more culturally responsive education (CRE). CRE is defined as, “a student-centred approach to teaching in which the students’ unique cultural strengths are identified and nurtured to promote student achievement and a sense of well-being about the student’s cultural place in the world” (Lynch, 2016, para. 2). Culturally responsive education provides students with the opportunity to use their own experiences and backgrounds to guide their learning.
Figure 6.1 TEDx talk by Isael Torres focused on cultural pedagogy and educational access and equality
Implications
The use of standardized tests to assess capabilities has worked against creating a cohesive, accessible curriculum. Instead of unifying STEM into a foundational way of thinking and creating a curriculum that honours it, STEM continues to be rather ambiguous in its definition, resulting in a heavy reliance on teaching the individual subjects. The tracking of students through a standardized test format has not resulted in an improvement for the US in global standings, nor has it unified the diverse groups of students that make up US schools. Rather, it has disaggregated students based on ethnicity, English language proficiency, socioeconomic status, and gender (Ansell, 2011; Miller, 2013). Failing to consider the diversity of thought and experience negates of the human side of the student population. Test scores are being used as the benchmark for policy decisions (Duignan & Nolen, 2019), which does not necessarily align with what is best for students. The use of standardized testing “has become entrenched in our society, and the collection of such data has exploded in its frequency, in its undue influence on the curriculum, and in its use for making life-impacting decisions about children, teachers, administrators, and schools” (Miller, 2013, para. 5). Foundational thinking and innovative teaching pedagogy provide the opportunity for CRE to be integrated into teaching practice. Teaching in a silo limits the capacity for divergent thinking and students’ cultural strengths to be utilized.
Connection of STEM Education in Teaching to Respect for Participant Autonomy and Independence
Though STEM education at the K-16 level has continued to expand, and numerous programs have been created to promote STEM with girls and women, there continues to be a gender gap in STEM fields and careers (Archer et al., 2012; Chang, 2019; UNESCO, 2017). Research has noted numerous reasons for this gap, not limited to gender stereotypes (Shapiro & Williams, 2011; UNESCO, 2017), earning discrepancies (Xu, 2015), teacher bias, and a lack of female voice in STEM programming. Figure 6.2 highlights the “ecological framework of factors influencing girls’ and women’s participation, achievement and progression in STEM studies” (p. 40).
When determining best practices for STEM education, one could consider a female perspective when designing a curriculum. From a social justice perspective, one could look at the recognitive justice of female representation in STEM education and instruction (Lambert, 2018). Is there an equal representation of females in textbooks, curriculum, etc., to ensure that girls feel a sense of community in their learning?
Currently, STEM education as a silo approach is predominantly taught from a male perspective, which limits many girls’ capacity to develop a STEM identity (Archer et al., 2012; Calabrese Barton & Brickhouse, 2006; Haussler & Hoffmann, 2002). Education requires a shift in how STEM classes are taught at a K-12 level. A foundational approach could be beneficial to aiding girls in developing their STEM identity (Baker, 2012; Haussler & Hoffmann, 2002; UNESCO, 2017). Research has shown that girls have responded favourably to a curriculum that incorporates “. . . a strong conceptual framework, [and is] contextualized and relevant to real-world situations” (UNESCO, 2017, p.67). Additionally, a foundational approach to STEM creates an environment that could build girls’ interests, as the curricula can provide a “varied experience, which integrates social and scientific issues, provides opportunities for genuine inquiry, involves real-world experience, as well as opportunities for experimentation, practice, reflection and conceptualization” (UNESCO, 2017, p. 67).
Ethical Considerations of a STEM Identity for Girls in a Silo Approach
From a consequentialist ethical perspective (Farrow, 2016), STEM education thus far has been known to limit autonomy and independence of women. By focusing on a silo approach, teachers and policymakers propagate pre-existing gender stereotypes. These stereotypes may contribute to expanding the gender gap in STEM-based fields and careers. When pedagogy focuses on a foundational way of thinking, thus prioritizing inquiry and real-world problems, girls demonstrate greater interest in pursuing STEM fields. Gender bias and stereotyping have limited women’s self-identity and their pursuit of STEM-based fields. These biases, through teachers, parents, and society as a whole, inadvertently, and at times intentionally, impact girls from a young age.
From a virtues perspective (Farrow, 2016) a shift in educational standards is required in order to meet the needs of all learners. The bias towards males as scientists and mathematicians is antiquated, and research has shown that gender plays little role in the learning of individual subjects. These biases cause harm to girls developing a sense of identity in the STEM subjects. Every student has a right to learn and to gain education free from stereotypes and bias.
Viewing women’s roles in STEM from a deontological ethical perspective (Farrow, 2016), women and men are equal neurologically. Research has demonstrated that gender bias and stereotypes are not founded in science and fact, as the human brain is designed in much the same way regardless of male or female genetics (UNESCO, 2017). Eradicating these stereotypes can lead to an equality of learning and provide more freedom for women to focus on STEM subjects. Providing equity and promoting women’s pursuit of STEM-focused programs could expand the depth and breadth of innovation.
Implications
Gender bias and stereotypes have greatly affected female STEM identities. These biases have limited their pursuit of STEM fields/careers. Research has shown that these stereotypes are not grounded in fact, but rather on assumptions made of male and female capabilities and skillsets (UNESCO, 2017).
STEM education requires shifting teaching pedagogy from a curriculum focused on the individual subjects or disciplines to one that incorporates inquiry, real-world problems, and a contextualized approach. Shifting the teaching of STEM can open up more opportunities for females to develop a stronger STEM identity. There is a need for recognitive justice in the STEM world. Women are woefully underrepresented in textbooks and learning materials. Through mentoring opportunities and more visibility of women in STEM, girls can begin to develop a better sense of their role and identity in the STEM subjects.
Conclusion
When considering a standardized approach for STEM education, it is vitally important to consider all of the stakeholders and their voices. When we teach STEM subjects as silos, many ethical issues arise that limit the efficacy of the learning, most notably for girls and minority groups.
When focusing on the T in STEM, students become trackable and reliant on technology for their learning. This tracking leads to an invasion of privacy and issues around consent. The reliance on technology for learning leads to more money being funnelled to tech giants, and further issues emerge regarding dependence on technology for achieving curricular outcomes.
The standardization of STEM into silos limits assessment and culturally responsive teaching. Any standardized assessment tool limits the diversity of thinking and culturally relevant curricula.
As there is a pronounced gap between women and men pursuing STEM careers, there is a limitation on diversity. Gender bias and stereotypes have negatively impacted female pursuit of STEM careers. It has led to a stereotype threat that propagates girls’ belief that they are just not as strong as boys in the STEM subjects.
STEM education is an important component of teaching practice. The focus of STEM as a curricular component should not be up for debate, only how it is to be defined. How it is taught should be of the biggest concern to all stakeholders. Those who are impacted the most by this ambiguity should be the ones whose voices are considered in this definition. Ensuring a program that is rich in diversity and that focuses on a foundational way of thinking could pave the way for equity and a voice for all.
Note: Using Farrow’s (2016) framework, Appendix A summarizes key ethical considerations when determining best practices for the integration of STEM education in K-12 schools. The impact of a misguided definition for STEM has numerous repercussions as noted in the framework.
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Seyranian, V., Madva, A., Duong, N., Abramzon, N., Tibbetts, Y., & Harackiewicz, J. M. (2018). The longitudinal effects of STEM identity and gender on flourishing and achievement in college physics. International Journal of STEM Education , 5 (1). https://doi.org/10.1186/s40594-018-0137-0
Shapiro, J. R. & Williams, A. M. (2012). The role of stereotype threats in undermining girls’ and women’s performance and interest in STEM fields. Sex Roles , 66 , 175–183. https://doi.org/10.1007/s11199-011-0051-0
Siekmann, G. & Korbel, P. (2016). Defining “STEM” skills: Review and synthesis of the literature. National Centre for Vocational Education Research. https://files.eric.ed.gov/fulltext/ED570655.pdf
Science from Scientists (n.d.). The STEM crisis & our solution. Science from Scientists . https://www.sciencefromscientists.org/the-stem-crisis
UNESCO. (2017). Cracking the code: Girls’ and women’s education in science, technology, engineering and mathematics (STEM) . United Nations Educational, Scientific and Cultural Organization. https://unesdoc.unesco.org/ark:/48223/pf0000253479
Villapaz, L. (2014, August 15). How Google took over the American classroom and is creating a Gmail generation. International Business Times. https://www.ibtimes.com/how-google-took-over-american-classroom-creating-gmail-generation-1657852
The White House. (2009, November 23). President Obama launches “Educate to Innovate” campaign for excellence in science, technology, engineering & math (stem) education [Press release]. https://obamawhitehouse.archives.gov/the-press-office/president-obama-launches-educate-innovate-campaign-excellence-science-technology-en
Williams, P. J. (2011). STEM education: Proceed with caution. Design and Technology Education , 16 (1), 26–35. https://eric.ed.gov/?id=EJ916494
Xu, Y. (2015). Focusing on women in STEM: A longitudinal examination of gender-based learning gap of college graduates. Journal of Higher Education , 86 (4), 489–523. https://doi.org/10.1353/jhe.2015.0020
Zakaria, F. (2015, March 26). Why America’s obsession with STEM education is dangerous. The Washington Post. https://www.washingtonpost.com/opinions/why-stem-wont-make-us-successful/2015/03/26/5f4604f2-d2a5-11e4-ab77-9646eea6a4c7_story.html
Zeide, E. (2017). The limits of education purpose limitations. University of Miami Law Review , 71 (2), 494–527. https://repository.law.miami.edu/umlr/vol71/iss2/8/
Appendix A
| Principle | Duties & Responsibilities (deontological) | Outcome (consequentialist) | Personal Development (Virtue) |
|---|---|---|---|
| Full disclosure |
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| Privacy, data security, & informed consent |
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| Avoiding harm & minimizing risk |
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| Autonomy & independence |
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Media Attributions
- Figure 1 Factors Impacting girls © Jennifer Ansorger is licensed under a CC BY (Attribution) license
- Figure 6.2: STEM Programs Designed for Girls © Jennifer Ansorger is licensed under a CC BY (Attribution) license


4.4: Derivation of the Mathematical Statement of the Second Law
4.4.1 The existence of the entropy function
This section derives the existence and properties of the state function called entropy.
Consider an arbitrary cyclic process of a closed system. To avoid confusion, this system will be the “experimental system” and the process will be the “experimental process” or “experimental cycle.” There are no restrictions on the contents of the experimental system—it may have any degree of complexity whatsoever. The experimental process may involve more than one kind of work, phase changes and reactions may occur, there may be temperature and pressure gradients, constraints and external fields may be present, and so on. All parts of the process must be either irreversible or reversible, but not impossible.
Figure 4.8
Experimental system, Carnot engine (represented by a small square box), and heat reservoir. The dashed lines indicate the boundary of the supersystem.
(a) Reversible heat transfer between heat reservoir and Carnot engine.
(b) Heat transfer between Carnot engine and experimental system.
The infinitesimal quantities \(\dq'\) and \(\dq\) are positive for transfer in the directions indicated by the arrows.
We imagine that the experimental cycle is carried out in a special way that allows us to apply the Kelvin–Planck statement of the second law. The heat transferred across the boundary of the experimental system in each infinitesimal path element of the cycle is exchanged with a hypothetical Carnot engine. The combination of the experimental system and the Carnot engine is a closed supersystem (see Fig. 4.8). In the surroundings of the supersystem is a heat reservoir of arbitrary constant temperature \(T\subs{res}\). By allowing the supersystem to exchange heat with only this single heat reservoir, we will be able to apply the Kelvin–Planck statement to a cycle of the supersystem.
This procedure is similar to ones described by A. B. Pippard ( Elements of Classical Thermodynamics for Advanced Students of Physics , Cambridge University Press, Cambridge, 1966, Chap. 4); C. J. Adkins ( Equilibrium Thermodynamics , 3rd edition, Cambridge University Press, Cambridge, 1983, Chap. 5); and Peter T. Landsberg ( Thermodynamics and Statistical Mechanics , Dover Publications, Inc., New York, 1990, p. 53).
We assume that we are able to control changes of the work coordinates of the experimental system from the surroundings of the supersystem. We are also able to control the Carnot engine from these surroundings, for example by moving the piston of a cylinder-and-piston device containing the working substance. Thus the energy transferred by work across the boundary of the experimental system, and the work required to operate the Carnot engine, is exchanged with the surroundings of the supersystem.
During each stage of the experimental process with nonzero heat, we allow the Carnot engine to undergo many infinitesimal Carnot cycles with infinitesimal quantities of heat and work. In one of the isothermal steps of each Carnot cycle, the Carnot engine is in thermal contact with the heat reservoir, as depicted in Fig. 4.8(a). In this step the Carnot engine has the same temperature as the heat reservoir, and reversibly exchanges heat \(\dq'\) with it. The sign convention is that \(\dq'\) is positive if heat is transferred in the direction of the arrow, from the heat reservoir to the Carnot engine.
In the other isothermal step of the Carnot cycle, the Carnot engine is in thermal contact with the experimental system at a portion of the system’s boundary. as depicted in Fig. 4.8(b). The Carnot engine now has the same temperature, \(T\bd\), as the experimental system at this part of the boundary, and exchanges heat with it. The heat \(\dq\) is positive if the transfer is into the experimental system.
The relation between temperatures and heats in the isothermal steps of a Carnot cycle is given by Eq. 4.3.15. From this relation we obtain, for one infinitesimal Carnot cycle, the relation \(T\bd/T\subs{res}=\dq/\dq'\), or \begin{equation} \dq'=T\subs{res}\frac{\dq}{T\bd} \tag{4.4.1} \end{equation}
After many infinitesimal Carnot cycles, the experimental cycle is complete, the experimental system has returned to its initial state, and the Carnot engine has returned to its initial state in thermal contact with the heat reservoir. Integration of Eq. 4.4.1 around the experimental cycle gives the net heat entering the supersystem during the process: \begin{gather} q'=T\subs{res} \tag{4.4.2} \oint\!\frac{\dq}{T\bd} \end{gather} The integration here is over each path element of the experimental process and over each surface element of the boundary of the experimental system.
Keep in mind that the value of the cyclic integral \(\oint\dq/T\bd\) depends only on the path of the experimental cycle, that this process can be reversible or irreversible, and that \(T\subs{res}\) is a positive constant.
In this experimental cycle, could the net heat \(q'\) transferred to the supersystem be positive? If so, the net work would be negative (to make the internal energy change zero) and the supersystem would have converted heat from a single heat reservoir completely into work, a process the Kelvin–Planck statement of the second law says is impossible. Therefore it is impossible for \(q'\) to be positive, and from Eq. 4.4.2 we obtain the relation \begin{gather} \s{\oint\!\frac{\dq}{T\bd}\leq 0} \tag{4.4.3} \cond{(cyclic process of a closed system)} \end{gather} This relation is known as the Clausius inequality . It is valid only if the integration is taken around a cyclic path in a direction with nothing but reversible and irreversible changes—the path must not include an impossible change, such as the reverse of an irreversible change. The Clausius inequality says that if a cyclic path meets this specification, it is impossible for the cyclic integral \(\oint(\dq/T\bd)\) to be positive.
If the entire experimental cycle is adiabatic (which is only possible if the process is reversible), the Carnot engine is not needed and Eq. 4.4.3 can be replaced by \(\oint(\dq/T\bd)=0\).
Next let us investigate a reversible nonadiabatic process of the closed experimental system. Starting with a particular equilibrium state A, we carry out a reversible process in which there is a net flow of heat into the system, and in which \(\dq\) is either positive or zero in each path element. The final state of this process is equilibrium state B. If each infinitesimal quantity of heat \(\dq\) is positive or zero during the process, then the integral \(\int_{\tx{A}}^{\tx{B}}(\dq/T\bd)\) must be positive. In this case the Clausius inequality tells us that if the system completes a cycle by returning from state B back to state A by a different path, the integral \(\int_{\tx{B}}^{\tx{A}}(\dq/T\bd)\) for this second path must be negative. Therefore the change B\(\ra\)A cannot be carried out by any adiabatic process.
Any reversible process can be carried out in reverse. Thus, by reversing the reversible nonadiabatic process, it is possible to change the state from B to A by a reversible process with a net flow of heat out of the system and with \(\dq\) either negative or zero in each element of the reverse path. In contrast, the absence of an adiabatic path from B to A means that it is impossible to carry out the change A\(\ra\)B by a reversible adiabatic process.
The general rule, then, is that whenever equilibrium state A of a closed system can be changed to equilibrium state B by a reversible process with finite “one-way” heat (i.e., the flow of heat is either entirely into the system or else entirely out of it), it is impossible for the system to change from either of these states to the other by a reversible adiabatic process.
A simple example will relate this rule to experience. We can increase the temperature of a liquid by allowing heat to flow reversibly into the liquid. It is impossible to duplicate this change of state by a reversible process without heat—that is, by using some kind of reversible work. The reason is that reversible work involves the change of a work coordinate that brings the system to a different final state. There is nothing in the rule that says we can’t increase the temperature irreversibly without heat, as we can for instance with stirring work.
States A and B can be arbitrarily close. We conclude that every equilibrium state of a closed system has other equilibrium states infinitesimally close to it that are inaccessible by a reversible adiabatic process . This is Carathéodory’s principle of adiabatic inaccessibility. (Constantin Carathéodory in 1909 combined this principle with a mathematical theorem\(—\)Carathéodory’s theorem\(—\)to deduce the existence of the entropy function. The derivation outlined here avoids the complexities of that mathematical treatment and leads to the same results.)
Next let us consider the reversible adiabatic processes that are possible. To carry out a reversible adiabatic process, starting at an initial equilibrium state, we use an adiabatic boundary and slowly vary one or more of the work coordinates. A certain final temperature will result. It is helpful in visualizing this process to think of an \(N\)-dimensional space in which each axis represents one of the \(N\) independent variables needed to describe an equilibrium state. A point in this space represents an equilibrium state, and the path of a reversible process can be represented as a curve in this space.
A suitable set of independent variables for equilibrium states of a closed system of uniform temperature consists of the temperature \(T\) and each of the work coordinates (Sec. 3.10). We can vary the work coordinates independently while keeping the boundary adiabatic, so the paths for possible reversible adiabatic processes can connect any arbitrary combinations of work coordinate values.
There is, however, the additional dimension of temperature in the \(N\)-dimensional space. Do the paths for possible reversible adiabatic processes, starting from a common initial point, lie in a volume in the \(N\)-dimensional space? Or do they fall on a surface described by \(T\) as a function of the work coordinates? If the paths lie in a volume, then every point in a volume element surrounding the initial point must be accessible from the initial point by a reversible adiabatic path. This accessibility is precisely what Carathéodory’s principle of adiabatic inaccessibility denies. Therefore, the paths for all possible reversible adiabatic processes with a common initial state must lie on a unique surface . This is an \((N-1)\)-dimensional hypersurface in the \(N\)-dimensional space, or a curve if \(N\) is \(2\). One of these surfaces or curves will be referred to as a reversible adiabatic surface .
Now consider the initial and final states of a reversible process with one-way heat (i.e., each nonzero infinitesimal quantity of heat \(\dq\) has the same sign). Since we have seen that it is impossible for there to be a reversible adiabatic path between these states, the points for these states must lie on different reversible adiabatic surfaces that do not intersect anywhere in the \(N\)-dimensional space. Consequently, there is an infinite number of nonintersecting reversible adiabatic surfaces filling the \(N\)-dimensional space. (To visualize this for \(N=3\), think of a flexed stack of paper sheets; each sheet represents a different reversible adiabatic surface in three-dimensional space.) A reversible, nonadiabatic process with one-way heat is represented by a path beginning at a point on one reversible adiabatic surface and ending at a point on a different surface. If \(q\) is positive, the final surface lies on one side of the initial surface, and if \(q\) is negative, the final surface is on the opposite side.
4.4.2 Using reversible processes to define the entropy
The existence of reversible adiabatic surfaces is the justification for defining a new state function \(S\), the entropy . \(S\) is specified to have the same value everywhere on one of these surfaces, and a different, unique value on each different surface. In other words, the reversible adiabatic surfaces are surfaces of constant entropy in the \(N\)-dimensional space. The fact that the surfaces fill this space without intersecting ensures that \(S\) is a state function for equilibrium states, because any point in this space represents an equilibrium state and also lies on a single reversible adiabatic surface with a definite value of \(S\).
Figure 4.9 A family of reversible adiabatic curves (two-dimensional reversible adiabatic surfaces) for an ideal gas with \(V\) and \(T\) as independent variables. A reversible adiabatic process moves the state of the system along a curve, whereas a reversible process with positive heat moves the state from one curve to another above and to the right. The curves are calculated for \(n = 1\mol\) and \(\CVm = (3/2)R\). Adjacent curves differ in entropy by \(1\units{J K\(^{-1}\)}\).
We know the entropy function must exist, because the reversible adiabatic surfaces exist. For instance, Fig. 4.9 shows a family of these surfaces for a closed system of a pure substance in a single phase. In this system, \(N\) is equal to 2, and the surfaces are two-dimensional curves. Each curve is a contour of constant \(S\). At this stage in the derivation, our assignment of values of \(S\) to the different curves is entirely arbitrary.
How can we assign a unique value of \(S\) to each reversible adiabatic surface? We can order the values by letting a reversible process with positive one-way heat, which moves the point for the state to a new surface, correspond to an increase in the value of \(S\). Negative one-way heat will then correspond to decreasing \(S\). We can assign an arbitrary value to the entropy on one particular reversible adiabatic surface. (The third law of thermodynamics is used for this purpose—see Sec. 6.1.) Then all that is needed to assign a value of \(S\) to each equilibrium state is a formula for evaluating the difference in the entropies of any two surfaces.
Figure 4.10
Reversible paths in \(V\)–\(T\) space. The thin curves are reversible adiabatic surfaces.
(a) Two paths connecting the same pair of reversible adiabatic surfaces.
(b) A cyclic path.
Consider a reversible process with positive one-way heat that changes the system from state A to state B. The path for this process must move the system from a reversible adiabatic surface of a certain entropy to a different surface of greater entropy. An example is the path A\(\ra\)B in Fig. 4.10(a). (The adiabatic surfaces in this figure are actually two-dimensional curves.) As before, we combine the experimental system with a Carnot engine to form a supersystem that exchanges heat with a single heat reservoir of constant temperature \(T\subs{res}\). The net heat entering the supersystem, found by integrating Eq. 4.4.1, is \begin{equation} q' = T\subs{res} \int_{\tx{A}}^{\tx{B}} \frac{\dq}{T\bd} \tag{4.4.4} \end{equation} and it is positive.
Suppose the same experimental system undergoes a second reversible process, not necessarily with one-way heat, along a different path connecting the same pair of reversible adiabatic surfaces. This could be path C\(\ra\)D in Fig. 4.10(a). The net heat entering the supersystem during this second process is \(q''\): \begin{equation} q'' = T\subs{res} \int_{\tx{C}}^{\tx{D}} \frac{\dq}{T\bd} \tag{4.4.5} \end{equation} We can then devise a cycle of the supersystem in which the experimental system undergoes the reversible path A\(\ra\)B\(\ra\)D\(\ra\)C\(\ra\)A, as shown in Fig. 4.10(b). Step A\(\ra\)B is the first process described above, step D\(\ra\)C is the reverse of the second process described above, and steps B\(\ra\)D and C\(\ra\)A are reversible and adiabatic. The net heat entering the supersystem in the cycle is \(q' - q''\). In the reverse cycle the net heat is \(q''-q'\). In both of these cycles the heat is exchanged with a single heat reservoir; therefore, according to the Kelvin–Planck statement, neither cycle can have positive net heat. Therefore \(q'\) and \(q''\) must be equal, and Eqs. 4.4.4 and 4.4.5 then show the integral \(\int(\dq/T\bd)\) has the same value when evaluated along either of the reversible paths from the lower to the higher entropy surface.
Note that since the second path (C\(\ra\)D) does not necessarily have one-way heat, it can take the experimental system through any sequence of intermediate entropy values, provided it starts at the lower entropy surface and ends at the higher. Furthermore, since the path is reversible, it can be carried out in reverse resulting in reversal of the signs of \(\Del S\) and \(\int(\dq/T\bd)\).
It should now be apparent that a satisfactory formula for defining the entropy change of a reversible process in a closed system is \begin{gather} \s{ \Del S = \int\!\frac{\dq}{T\bd} } \tag{4.4.6} \cond{(reversible process,} \nextcond{closed system)} \end{gather} This formula satisfies the necessary requirements: it makes the value of \(\Del S\) positive if the process has positive one-way heat, negative if the process has negative one-way heat, and zero if the process is adiabatic. It gives the same value of \(\Del S\) for any reversible change between the same two reversible adiabatic surfaces, and it makes the sum of the \(\Del S\) values of several consecutive reversible processes equal to \(\Del S\) for the overall process.
In Eq. 4.4.6, \(\Del S\) is the entropy change when the system changes from one arbitrary equilibrium state to another. If the change is an infinitesimal path element of a reversible process, the equation becomes \begin{gather} \s{ \dif S = \frac{\dq}{T\bd} } \tag{4.4.7} \cond{(reversible process,} \nextcond{closed system)} \end{gather} It is common to see this equation written in the form \(\dif S = \dq\rev/T\), where \(\dq\rev\) denotes an infinitesimal quantity of heat in a reversible process.
In Eq. 4.4.7, the quantity \(1/T\bd\) is called an integrating factor for \(\dq\), a factor that makes the product \((1/T\bd)\dq\) be the infinitesimal change of a state function. The quantity \(c/T\bd\), where \(c\) is any nonzero constant, would also be a satisfactory integrating factor; so the definition of entropy, using \(c{=}1\), is actually one of an infinite number of possible choices for assigning values to the reversible adiabatic surfaces.
4.4.3 Some properties of the entropy
It is not difficult to show that the entropy of a closed system in an equilibrium state is an extensive property. Suppose a system of uniform temperature \(T\) is divided into two closed subsystems A and B. When a reversible infinitesimal change occurs, the entropy changes of the subsystems are \(\dif S\subs{A} = \dq\subs{A}/T\) and \(\dif S\subs{B} = \dq\subs{B}/T\) and of the system \(\dif S = \dq/T\). But \(\dq\) is the sum of \(\dq\subs{A}\) and \(\dq\subs{B}\), which gives \(\dif S = \dif S\subs{A} + \dif S\subs{B}\). Thus, the entropy changes are additive, so that entropy must be extensive: \(S=S\subs{A}+S\subs{B}\). (The argument is not quite complete, because we have not shown that when each subsystem has an entropy of zero, so does the entire system. The zero of entropy will be discussed in Sec. 6.1.)
How can we evaluate the entropy of a particular equilibrium state of the system? We must assign an arbitrary value to one state and then evaluate the entropy change along a reversible path from this state to the state of interest using \(\Del S=\int(\dq/T\bd)\).
We may need to evaluate the entropy of a non equilibrium state. To do this, we imagine imposing hypothetical internal constraints that change the nonequilibrium state to a constrained equilibrium state with the same internal structure. Some examples of such internal constraints were given in Sec. 2.4.4, and include rigid adiabatic partitions between phases of different temperature and pressure, semipermeable membranes to prevent transfer of certain species between adjacent phases, and inhibitors to prevent chemical reactions.
We assume that we can, in principle, impose or remove such constraints reversibly without heat, so there is no entropy change. If the nonequilibrium state includes macroscopic internal motion, the imposition of internal constraints involves negative reversible work to bring moving regions of the system to rest. This concept amounts to defining the entropy of a state with macroscopic internal motion to be the same as the entropy of a state with the same internal structure but without the motion, i.e., the same state frozen in time. By this definition, \(\Del S\) for a purely mechanical process (Sec. 3.2.3) is zero.
If the system is nonuniform over its extent, the internal constraints will partition it into practically-uniform regions whose entropy is additive. The entropy of the nonequilibrium state is then found from \(\Del S=\int(\dq/T\bd)\) using a reversible path that changes the system from an equilibrium state of known entropy to the constrained equilibrium state with the same entropy as the state of interest. This procedure allows every possible state (at least conceptually) to have a definite value of \(S\).


1.1: What is Geology?
In its broadest sense, geology is the study of Earth—its interior and its exterior surface, the minerals, rocks and other materials that are around us, the processes that have resulted in the formation of those materials, the water that flows over the surface and through the ground, the changes that have taken place over the vastness of geological time, and the changes that we can anticipate will take place in the near future. Geology is a science, meaning that we use deductive reasoning and scientific methods to understand geological problems. It is, arguably, the most integrated of all of the sciences because it involves the understanding and application of all of the other sciences: physics, chemistry, biology, mathematics, astronomy, and others. But unlike most of the other sciences, geology has an extra dimension, that of time—deep time—billions of years of it. Geologists study the evidence that they see around them, but in most cases, they are observing the results of processes that happened thousands, millions, and even billions of years in the past. Those were processes that took place at incredibly slow rates—millimeters per year to centimeters per year—but because of the amount of time available, they produced massive results.
Geology is displayed on a grand scale in mountainous regions, perhaps nowhere better than the Rocky Mountains in Canada (Figure \(\PageIndex{1}\)). The peak on the right is Rearguard Mountain, which is a few kilometers northeast of Mount Robson, the tallest peak in the Canadian Rockies (3,954 meters). The large glacier in the middle of the photo is the Robson Glacier. The river flowing from Robson Glacier drains into Berg Lake in the bottom right. There are many geological features portrayed here. The sedimentary rock that these mountains are made of formed in ocean water over 500 million years ago. A few hundred million years later, these beds were pushed east for tens to hundreds of kilometers by tectonic plate convergence and also pushed up to thousands of meters above sea level. Over the past two million years this area—like most of the rest of Canada—has been repeatedly glaciated, and the erosional effects of those glaciations are obvious.
The Robson Glacier is now only a small remnant of its size during the Little Ice Age of the 15th to 18th centuries, and even a lot smaller that it was just over a century ago in 1908. The distinctive line on the slope on the left side of both photos shows the elevation of the edge of the glacier a few hundred years ago. Like almost all other glaciers in the world, it receded after the 18th century because of natural climate change, is now receding even more rapidly because of human-caused climate change.
Geology is also about understanding the evolution of life on Earth; about discovering resources such as water, metals and energy; about recognizing and minimizing the environmental implications of our use of those resources; and about learning how to mitigate the hazards related to earthquakes, volcanic eruptions, and slope failures. All of these aspects of geology, and many more, are covered in this textbook.
What are scientific methods?
There is no single method of inquiry that is specifically the “scientific method”; furthermore, scientific inquiry is not necessarily different from serious research in other disciplines. The most important thing that those involved in any type of inquiry must do is to be skeptical. As the physicist Richard Feynman once said: the first principle of science is that “you must not fool yourself—and you are the easiest person to fool.” A key feature of serious inquiry is the creation of a hypothesis (a tentative explanation) that could explain the observations that have been made, and then the formulation and testing (by experimentation) of one or more predictions that follow from that hypothesis.
For example, we might observe that most of the cobbles in a stream bed are well rounded (see photo above), and then derive the hypothesis that the rocks are rounded by transportation along the stream bed. A prediction that follows from this hypothesis is that cobbles present in a stream will become increasingly rounded as they are transported downstream. An experiment to test this prediction would be to place some angular cobbles in a stream, label them so that we can be sure to find them again later, and then return at various time intervals (over a period of years) to carefully measure their locations and roundness.
A critical feature of a good hypothesis and any resulting predictions is that they must be testable. For example, an alternative hypothesis to the one above is that an extraterrestrial organization creates rounded cobbles and places them in streams when nobody is looking. This may indeed be the case, but there is no practical way to test this hypothesis. Most importantly, there is no way to prove that it is false, because if we aren’t able to catch the aliens at work, we still won’t know if they did it!
Media Attributions
- Figure \(\PageIndex{1}\) (left): © Steven Earle. CC BY.
- Figure \(\PageIndex{1}\) (right): A.P. Coleman. Public domain. Source: Arthur P. Coleman Collection at Victoria University Library.
- Figure \(\PageIndex{2}\): © Steven Earle. CC BY.


2.1: Introduction to Functions of Multiple Variables
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In Introduction to Applications of Derivatives, we studied how to determine the maximum and minimum of a function of one variable over a closed interval. This function might represent the temperature over a given time interval, the position of a car as a function of time, or the altitude of a jet plane as it travels from New York to San Francisco. In each of these examples, the function has one independent variable.
Suppose, however, that we have a quantity that depends on more than one variable. For example, temperature can depend on location and the time of day, or a company’s profit model might depend on the number of units sold and the amount of money spent on advertising. In this chapter, we look at a company that produces golf balls. We develop a profit model and, under various restrictions, we find that the optimal level of production and advertising dollars spent determines the maximum possible profit. Depending on the nature of the restrictions, both the method of solution and the solution itself changes.
When dealing with a function of more than one independent variable, several questions naturally arise. For example, how do we calculate limits of functions of more than one variable? The definition of derivative we used before involved a limit. Does the new definition of derivative involve limits as well? Do the rules of differentiation apply in this context? Can we find relative extrema of functions using derivatives? All these questions are answered in this chapter.


2.3: Problem Solving Strategies
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Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.
Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
How to Solve It
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]
George Pólya, circa 1973
- Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵
In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:
- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?
You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.
This brings us to the most important problem solving strategy of all:
Problem Solving Strategy 2 (Try Something!).
If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.
And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.
(Payback)
Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?
Think/Pair/Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.
Problem Solving Strategy 3 (Draw a Picture).
Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?
Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?
After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.
Problem Solving Strategy 4 (Make Up Numbers).
Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!
You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.
Watch the solution only after you tried this strategy for yourself.
If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!
(Squares on a Chess Board)
How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)
Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?
It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”
Problem Solving Strategy 5 (Try a Simpler Problem).
Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?
Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).
Problem Solving Strategy 6 (Work Systematically).
If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.
For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:
| size of board | # of 1 × 1 squares | # of 2 × 2 squares | # of 3 × 3 squares | # of 4 × 4 squares | … |
| 1 by 1 | 1 | 0 | 0 | 0 | |
| 2 by 2 | 4 | 1 | 0 | 0 | |
| 3 by 3 | 9 | 4 | 1 | 0 | |
| … |
Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).
Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!
For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.
Problem Solving Strategy 8 (Look for and Explain Patterns).
Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.
Think / Pair / Share
If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:
- Describe all of the patterns you see in the table.
- Can you explain and justify any of the patterns you see? How can you be sure they will continue?
- What calculation would you do to find the total number of squares on a 100 × 100 chess board?
(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)
(Broken Clock)
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)
Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?
Problem Solving Strategy 9 (Find the Math, Remove the Context).
Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:
- What is the sum of all the numbers on the clock’s face?
- Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
- How do I know when I am done? When should I stop looking?
Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.
Problem Solving Strategy 10 (Check Your Assumptions).
When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?
In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:
Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.


<IP_ADDRESS>: Column Space
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We begin with the simple geometric interpretation of matrix-vector multiplication. Namely, the multiplication of the n-by-1 vector \(x\) by the m-by-n matrix \(A\) produces a linear combination of the columns of A. More precisely, if \(a_{j}\) denotes the jth column of A then
\[Ax = \begin{pmatrix} {a_{1}}&{a_{2}}&{\cdots}&{a_{n}} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}}\\ {\cdots}\\ {x_{n}} \end{pmatrix} \nonumber\]
\[= x_{1}a_{1}+x_{2}a_{2}+\cdots+x_{n}a_{n} \nonumber\]
The picture that I wish to place in your mind's eye is that AxAx lies in the subspace spanned by the columns of \(A\). This subspace occurs so frequently that we find it useful to distinguish it with a definition.
Column Space
The column space of the m-by-n matrix \(S\) is simply the span of the its columns, i.e. \(Ra(S) \equiv \{Sx | x \in \mathbb{R}^{n}\}\) subspace of \(\mathcal{R}^{m}\) stands for range in this context.The notation \(R_{a}\) stands for range in this context.
Example
Let us examine the matrix:
\[A = \begin{pmatrix} {0}&{1}&{0}&{0}\\ {-1}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]
The column space of this matrix is:
\[Ra(A) = \{x_{1} \begin{pmatrix} {0}\\ {-1}\\ {0} \end{pmatrix}+x_{2} \begin{pmatrix} {1}\\ {0}\\ {0} \end{pmatrix}+x_{3} \begin{pmatrix} {0}\\ {1}\\ {0} \end{pmatrix}+x_{4} \begin{pmatrix} {0}\\ {0}\\ {1} \end{pmatrix} | x \in \mathbb{R}^{4}\} \nonumber\]
As the third column is simply a multiple of the first, we may write:
\[Ra(A) = \{x_{1} \begin{pmatrix} {0}\\ {1}\\ {0} \end{pmatrix}+x_{2} \begin{pmatrix} {1}\\ {0}\\ {0} \end{pmatrix}+x_{3} \begin{pmatrix} {0}\\ {0}\\ {1} \end{pmatrix} | x \in \mathbb{R}^{3}\} \nonumber\]
As the three remaining columns are linearly independent we may go no further. In this case, \(Ra(A)\) comprises all of \(\mathbb{R}^{3}\)
Method for Finding a Basis
To determine the basis for \(Ra(A)\) (where \(A\) is an arbitrary matrix) we must find a way to discard its dependent columns. In the example above, it was easy to see that columns 1 and 3 were colinear. We seek, of course, a more systematic means of uncovering these, and perhaps other less obvious, dependencies. Such dependencies are more easily discerned from the row reduced form. In the reduction of the above problem, we come very easily to the matrix
\[A_{red} = \begin{pmatrix} {-1}&{0}&{1}&{0}\\ {0}&{1}&{0}&{0}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]
Once we have done this, we can recognize that the pivot column are the linearly independent columns of \(A_{red}\). One now asks how this might help us distinguish the independent columns of A. For, although the rows of \(A_{red}\) are linear combinations of the rows of \(A\) pay attention to the indices of the pivot columns . In our example, columns \(\{1, 2, 4\}\) are the pivot columns of \(A_{red}\) and hence the first, second, and fourth columns of \(A\) i.e.,
\[\{\begin{pmatrix} {0}\\ {-1}\\ {0} \end{pmatrix}, \begin{pmatrix} {1}\\ {0}\\ {0} \end{pmatrix}, \begin{pmatrix} {0}\\ {0}\\ {1} \end{pmatrix}\} \nonumber\]
comprise a basis for \(Ra(A)\):
Definition: A Basis for the Column Space
Suppose \(A\) is m-by-n. If columns \(\{c_{j} | j = 1, \cdots, r\}\) are the pivot columns of \(A_{red}\) then columns \(\{c_{j} | j = 1, \cdots, r\}\) of \(A\) constitute a basis for \(Ra(A)\)


3.5: Sum-to-Product and Product-to-Sum Formulas
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Learning Objectives
- Express products as sums.
- Express sums as products.
A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions.
Figure \(\PageIndex{1}\): The UCLA marching band (credit: Eric Chan, Flickr).
For example, Figure \(\PageIndex{2}\) represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.
Figure \(\PageIndex{2}\)
Expressing Products as Sums
We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.
Expressing Products as Sums for Cosine
We can derive the product-to-sum formula from the sum and difference identities for cosine . If we add the two equations, we get:
\[\begin{align*} \cos \alpha \cos \beta+\sin \alpha \sin \beta&= \cos(\alpha-\beta)\\[4pt] \underline{+ \cos \alpha \cos \beta-\sin \alpha \sin \beta}&= \underline{ \cos(\alpha+\beta) }\\[4pt] 2 \cos \alpha \cos \beta&= \cos(\alpha-\beta)+\cos(\alpha+\beta)\end{align*}\]
Then, we divide by 2 to isolate the product of cosines:
\[ \cos \alpha \cos \beta= \dfrac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)] \label{eq1}\]
How to: Given a product of cosines, express as a sum
- Write the formula for the product of cosines.
- Substitute the given angles into the formula.
- Simplify.
Example \(\PageIndex{1}\): Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine
Write the following product of cosines as a sum: \(2\cos\left(\dfrac{7x}{2}\right) \cos\left(\dfrac{3x}{2}\right)\).
Solution
We begin by writing the formula for the product of cosines (Equation \ref{eq1}):
\[ \cos \alpha \cos \beta = \dfrac{1}{2}[ \cos(\alpha-\beta)+\cos(\alpha+\beta) ] \nonumber \]
We can then substitute the given angles into the formula and simplify.
\[\begin{align*} 2 \cos\left(\dfrac{7x}{2}\right)\cos\left(\dfrac{3x}{2}\right)&= 2\left(\dfrac{1}{2}\right)[ \cos\left(\dfrac{7x}{2}-\dfrac{3x}{2}\right)+\cos\left(\dfrac{7x}{2}+\dfrac{3x}{2}\right) ]\\[4pt] &= \cos\left(\dfrac{4x}{2}\right)+\cos\left(\dfrac{10x}{2}\right) \\[4pt] &= \cos 2x+\cos 5x \end{align*}\]
Exercise \(\PageIndex{1}\)
Use the product-to-sum formula (Equation \ref{eq1}) to write the product as a sum or difference: \(\cos(2\theta)\cos(4\theta)\).
- Answer
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\(\dfrac{1}{2}(\cos 6\theta+\cos 2\theta)\)
Expressing the Product of Sine and Cosine as a Sum
Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine . If we add the sum and difference identities, we get:
\[\begin{align*} \cos \alpha \cos \beta+\sin \alpha \sin \beta&= \cos(\alpha-\beta)\\[4pt] \underline{+ \cos \alpha \cos \beta-\sin \alpha \sin \beta}&= \cos(\alpha+\beta)\\[4pt] 2 \cos \alpha \cos \beta&= \cos(\alpha-\beta)+\cos(\alpha+\beta)\\[4pt] \text{Then, we divide by 2 to isolate the product of cosines:}\\[4pt] \cos \alpha \cos \beta&= \dfrac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha+\beta)\right] \end{align*}\]
Example \(\PageIndex{2}\): Writing the Product as a Sum Containing only Sine or Cosine
Express the following product as a sum containing only sine or cosine and no products: \(\sin(4\theta)\cos(2\theta)\).
Solution
Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.
\[\begin{align*} \sin \alpha \cos \beta&= \dfrac{1}{2}[ \sin(\alpha+\beta)+\sin(\alpha-\beta) ]\\[4pt] \sin(4\theta)\cos(2\theta)&= \dfrac{1}{2}[\sin(4\theta+2\theta)+\sin(4\theta-2\theta)]\\[4pt] &= \dfrac{1}{2}[\sin(6\theta)+\sin(2\theta)] \end{align*}\]
Exercise \(\PageIndex{2}\)
Use the product-to-sum formula to write the product as a sum: \(\sin(x+y)\cos(x−y)\).
- Answer
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\(\dfrac{1}{2}(\sin 2x+\sin 2y)\)
Expressing Products of Sines in Terms of Cosine
Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:
\[\begin{align*} \cos(\alpha-\beta)&= \cos \alpha \cos \beta+\sin \alpha \sin \beta\\[4pt] \underline{-\cos(\alpha+\beta)}&= -(\cos \alpha \cos \beta-\sin \alpha \sin \beta)\\[4pt] \cos(\alpha-\beta)-\cos(\alpha+\beta)&= 2 \sin \alpha \sin \beta\\[4pt] \text{Then, we divide by 2 to isolate the product of sines:}\\[4pt] \sin \alpha \sin \beta&= \dfrac{1}{2}[ \cos(\alpha-\beta)-\cos(\alpha+\beta) ] \end{align*}\]
Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.
THE PRODUCT-TO-SUM FORMULAS
The product-to-sum formulas are as follows:
\[\cos \alpha \cos \beta=\dfrac{1}{2}[\cos(\alpha−\beta)+\cos(\alpha+\beta)]\]
\[\sin \alpha \cos \beta=\dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha−\beta)]\]
\[\sin \alpha \sin \beta=\dfrac{1}{2}[\cos(\alpha−\beta)−\cos(\alpha+\beta)]\]
\[\cos \alpha \sin \beta=\dfrac{1}{2}[\sin(\alpha+\beta)−\sin(\alpha−\beta)]\]
Example \(\PageIndex{3}\): Express the Product as a Sum or Difference
Write \(\cos(3\theta) \cos(5\theta)\) as a sum or difference.
Solution
We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.
\[\begin{align*} \cos \alpha \cos \beta&= \dfrac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]\\[4pt] \cos(3\theta)\cos(5\theta)&= \dfrac{1}{2}[\cos(3\theta-5\theta)+\cos(3\theta+5\theta)]\\[4pt] &= \dfrac{1}{2}[\cos(2\theta)+\cos(8\theta)]\qquad \text{Use even-odd identity} \end{align*}\]
Exercise \(\PageIndex{3}\)
Use the product-to-sum formula to evaluate \(\cos \dfrac{11\pi}{12} \cos \dfrac{\pi}{12}\).
- Answer
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\(\dfrac{−2−\sqrt{3}}{4}\)
Expressing Sums as Products
Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine . Let \(\dfrac{u+v}{2}=\alpha\) and \(\dfrac{u−v}{2}=\beta\).
Then,
\[\begin{align*} \alpha+\beta&= \dfrac{u+v}{2}+\dfrac{u-v}{2}\\[4pt] &= \dfrac{2u}{2}\\[4pt] &= u \end{align*}\]
\[\begin{align*} \alpha-\beta&= \dfrac{u+v}{2}-\dfrac{u-v}{2}\\[4pt] &= \dfrac{2v}{2}\\[4pt] &= v \end{align*}\]
Thus, replacing \(\alpha\) and \(\beta\) in the product-to-sum formula with the substitute expressions, we have
\[\begin{align*} \sin \alpha \cos \beta&= \dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\\[4pt] \sin \left ( \frac{u+v}{2} \right ) \cos \left ( \frac{u-v}{2} \right )&= \frac{1}{2}[\sin u + \sin v]\qquad \text{Substitute for } (\alpha+\beta) \text{ and } (\alpha\beta)\\[4pt] 2\sin\left(\dfrac{u+v}{2}\right) \cos\left(\dfrac{u-v}{2}\right)&= \sin u+\sin v \end{align*}\]
The other sum-to-product identities are derived similarly.
SUM-TO-PRODUCT FORMULAS
The sum-to-product formulas are as follows:
\[\sin \alpha+\sin \beta=2\sin\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha−\beta}{2}\right)\]
\[\sin \alpha-\sin \beta=2\sin\left(\dfrac{\alpha-\beta}{2}\right)\cos\left(\dfrac{\alpha+\beta}{2}\right)\]
\[\cos \alpha−\cos \beta=−2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha−\beta}{2}\right)\]
\[\cos \alpha+\cos \beta=2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha−\beta}{2}\right)\]
Example \(\PageIndex{4}\): Writing the Difference of Sines as a Product
Write the following difference of sines expression as a product: \(\sin(4\theta)−\sin(2\theta)\).
Solution
We begin by writing the formula for the difference of sines.
\[\begin{align*} \sin \alpha-\sin \beta&= 2\sin\left(\dfrac{\alpha-\beta}{2}\right)\cos\left(\dfrac{\alpha+\beta}{2}\right)\\[4pt] \text {Substitute the values into the formula, and simplify.}\\[4pt] \sin(4\theta)-\sin(2\theta)&= 2\sin\left(\dfrac{4\theta-2\theta}{2}\right) \cos\left(\dfrac{4\theta+2\theta}{2}\right)\\[4pt] &= 2\sin\left(\dfrac{2\theta}{2}\right) \cos\left(\dfrac{6\theta}{2}\right)\\[4pt] &= 2 \sin \theta \cos(3\theta) \end{align*}\]
Exercise \(\PageIndex{4}\)
Use the sum-to-product formula to write the sum as a product: \(\sin(3\theta)+\sin(\theta)\).
- Answer
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\(2\sin(2\theta)\cos(\theta)\)
Example \(\PageIndex{5}\): Evaluating Using the Sum-to-Product Formula
Evaluate \(\cos(15°)−\cos(75°)\). Check the answer with a graphing calculator.
Solution
We begin by writing the formula for the difference of cosines.
\[\begin{align*}
\cos \alpha-\cos \beta&= -2 \sin\left(\dfrac{\alpha+\beta}{2}\right) \sin\left(\dfrac{\alpha-\beta}{2}\right)\\[4pt]
\text {Then we substitute the given angles and simplify.}\\[4pt]
\cos(15^{\circ})-\cos(75^{\circ})&= -2\sin\left(\dfrac{15^{\circ}+75^{\circ}}{2}\right) \sin\left(\dfrac{15^{\circ}-75^{\circ}}{2}\right)\\[4pt]
&= -2\sin(45^{\circ}) \sin(-30^{\circ})\\[4pt]
&= -2\left(\dfrac{\sqrt{2}}{2}\right)\left(-\dfrac{1}{2}\right)\\[4pt]
&= \dfrac{\sqrt{2}}{2}
\end{align*}\]
Example \(\PageIndex{6}\): Proving an Identity
Prove the identity:
\[\dfrac{\cos(4t)−\cos(2t)}{\sin(4t)+\sin(2t)}=−\tan t\]
Solution
We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.
\[\begin{align*} \dfrac{\cos(4t)-\cos(2t)}{\sin(4t)+\sin(2t)}&= \dfrac{-2 \sin\left(\dfrac{4t+2t}{2}\right) \sin\left(\dfrac{4t-2t}{2}\right)}{2 \sin\left(\dfrac{4t+2t}{2}\right) \cos\left(\dfrac{4t-2t}{2}\right)}\\[4pt] &= \dfrac{-2 \sin(3t)\sin t}{2 \sin(3t)\cos t}\\[4pt] &= -\dfrac{\sin t}{\cos t}\\[4pt] &= -\tan t \end{align*}\]
Analysis
Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.
Example \(\PageIndex{7}\): Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities
Verify the identity \({\csc}^2 \theta−2=\cos(2\theta)\sin2\theta\).
Solution
For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.
\[\begin{align*} \cos(2\theta)\sin2\theta&= \dfrac{1-2 {\sin}^2 \theta}{{\sin}^2 \theta}\\[4pt] &= \dfrac{1}{{\sin}^2 \theta}-\dfrac{2 {\sin}^2 \theta}{{\sin}^2 \theta}\\[4pt] &= {\csc}^2 \theta - 2 \end{align*}\]
Exercise \(\PageIndex{5}\)
Verify the identity \(\tan \theta \cot \theta−{\cos}^2 \theta={\sin}^2 \theta\).
- Answer
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\[\begin{align*} \tan \theta \cot \theta-{\cos}^2 \theta&= \left(\dfrac{\sin \theta}{\cos \theta}\right)\left(\dfrac{\cos \theta}{\sin \theta}\right)-{\cos}^2 \theta\\[4pt] &= 1-{\cos}^2 \theta\\[4pt] &= {\sin}^2 \theta \end{align*}\]
Media
Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.
Key Equations
Product-to-sum Formulas
\[\cos \alpha \cos \beta=\dfrac{1}{2}[\cos(\alpha−\beta)+\cos(\alpha+\beta)] \nonumber \]
\[\sin \alpha \cos \beta=\dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha−\beta)] \nonumber \]
\[\sin \alpha \sin \beta=\dfrac{1}{2}[\cos(\alpha−\beta)−\cos(\alpha+\beta)] \nonumber \]
\[\cos \alpha \sin \beta=\dfrac{1}{2}[\sin(\alpha+\beta)−\sin(\alpha−\beta)] \nonumber \]
Sum-to-product Formulas
\[\sin \alpha+\sin \beta=2\sin(\dfrac{\alpha+\beta}{2})\cos(\dfrac{\alpha−\beta}{2}) \nonumber \]
\[\sin \alpha-\sin \beta=2\sin(\dfrac{\alpha-\beta}{2})\cos(\dfrac{\alpha+\beta}{2}) \nonumber \]
\[\cos \alpha−\cos \beta=−2\sin(\dfrac{\alpha+\beta}{2})\sin(\dfrac{\alpha−\beta}{2}) \nonumber \]
\[\cos \alpha+\cos \beta=2\sin(\dfrac{\alpha+\beta}{2})\sin(\dfrac{\alpha−\beta}{2}) \nonumber \]
Key Concepts
- From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
- We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See Example \(\PageIndex{1}\), Example \(\PageIndex{2}\), and Example \(\PageIndex{3}\).
- We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
- We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See Example \(\PageIndex{4}\).
- Trigonometric expressions are often simpler to evaluate using the formulas. See Example \(\PageIndex{5}\).
- The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).