This unit bundles student expectations that address using trigonometric identities to simplify trigonometric expressions and processes to solve trigonometric equations. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.
Prior to this unit, in Geometry Unit 05, students determined the lengths of sides and the measures of angles in a right triangle by applying special right triangle relationships, the Pythagorean theorem, Pythagorean triples, and the trigonometric ratios, sine, cosine, and tangent. In Algebra II Units 04 and 08, students performed operations (including addition, subtraction, multiplication, and division) on polynomials and rational expressions. In Algebra II Units 02, 05 – 10, students formulated and solved absolute value, quadratic, square root, cubic, cube root, rational, exponential, and logarithmic equations. In Precalculus Units 03 – 05, students solved polynomial, power, rational, exponential, and logarithmic equations. In Precalculus Unit 07, students determined the values of the trigonometric functions at special angles in mathematical and real-world problem situations and used trigonometric ratios to solve problems. In Precalculus Unit 08, students developed and used sinusoidal functions to model situations and graphed trigonometric functions and their transformations.
During this unit, students analyze the symmetries and transformations of sine and cosine graphs to develop the even/odd identities and the cofunction identities for sine and cosine. Students use even/odd, cofunction, and reciprocal identities to simplify trigonometric expressions, using algebraic skills of simplifying fractions, combining like terms, distributing, and substituting equivalent expressions. Students use the quotient and Pythagorean identities to simplify additional trigonometric expressions, using algebra skills of simplifying fractions, combining like terms, distributing, factoring, combining rational expressions with common denominators, and substituting equivalent expressions. Students use the sum and difference identities for cosine and sine to simplify trigonometric expressions, using algebra skills of simplifying fractions, combining like terms, distributing, factoring, combining rational expressions with common denominators, and substituting equivalent expressions. Students solve trigonometric equations using graphs and tables and develop the general solutions to sine and cosine equations. Students use algebra skills and inverse trigonometric functions to solve sine and cosine trigonometric equations in mathematical problem situations. Students write general solutions to these equations and determine solutions within a given interval. Students generate trigonometric equations in mathematical and real-world problem situations and solve these equations using algebraic methods and inverse trigonometric functions.
After this unit, in Precalculus Units 10 – 12, students will continue to simplify trigonometric expressions and solve trigonometric equations when studying vectors, parametric equations, and polar equations. In subsequent mathematics courses, students will continue to apply these concepts as they arise in problem situations.
Algebraic manipulation serves an integral role in college readiness. Recognizing and using algebraic properties and procedures to transform expressions and solve equations is emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning B1, C1; III. Geometric Reasoning A3; VII. Functions B1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.
According to research, recognizing equivalent expressions and knowing how to simplify expressions are essential skills for solving algebraic problems (Van De Walle, Karp, & Bay-Williams, 2013). More specifically, “Students need an understanding of how to apply mathematical properties and how to preserve equivalence as they simplify” (Van De Walle et al., 2013, p. 268). The National Council of Teachers of Mathematics (2000) supports this emphasis on algebraic properties, abstraction, and structure by advocating that all high school students should understand the meaning of equivalent forms of expressions and equations and be able to write equivalent forms of equations and solve them with fluency. Research from Lloyd, Herbel-Eisenmann, & Star (2011) offers two additional insights regarding expressions and equations: first, that understanding of expressions is essential to a good foundation in algebra since expressions are the building blocks for equations and functions; and, second, that general, broadly applicable algorithms exist for solving many types of equations. Regarding trigonometric expressions and equations, Ellis, Bieda, & Knuth (2012) contend that engaging in verification and proof activities can provide new insights and allow students to make connections between various representations of mathematics, such as between graphs of trigonometric functions and unit-circle representations. Sinclair, Pimm, & Skelin (2012) add that familiarity with geometric concepts such as similar triangles and invariant length ratio properties contribute important insights into high school trigonometry courses, including sense-making about various trigonometric identities. The study of trigonometry is grounded in geometric thinking, including diagramming, constructing, and focusing on invariance (Sinclair et al., 2012). In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve analysis and understanding of elementary functions, including trigonometric and inverse trigonometric functions. Specifically, students must be familiar with the properties, algebra, graphs, and language of trigonometric functions, while also knowing the values of the trigonometric functions at multiples of , and .
College Board. (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
Ellis, A. B., Bieda, K., & Knuth, E. (2012). Developing essential understanding of proof and proving for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Lloyd, G. M., Herbel-Eisenmann, B., & Star, J. R.. (2010). Developing essential understanding of expressions, equations, and function for teaching mathematics in grades 6-8. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing essential understanding of geometry for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf.
Van De Walle, J. A., Karp, K. S., & Bay-Williams, J.M. (2013). Elementary and middle-school mathematics: Teaching developmentally. New York, NY: Pearson.