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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 09: Trigonometric Equations and Identities SUGGESTED DURATION : 10 days

#### Unit Overview

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 .

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.
Van De Walle, J. A., Karp, K. S., & Bay-Williams, J.M. (2013). Elementary and middle-school mathematics: Teaching developmentally. New York, NY: Pearson.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic expressions (numbers, variables, and operational symbols) are the basic tools of algebra.

• Why are algebraic expressions the basic tools of algebra?
• How are algebraic expressions used to express mathematical ideas and model mathematical and real-world situations?
• What operations do algebraic expressions undergo?
• How can two expressions be related?
• Why are algebraic expressions evaluated?

Equations can model problem situations and be solved using various methods.

• Why are equations used to model problem situations?
• How are equations used to model problem situations?
• What methods can be used to solve equations?
• Why is it essential to solve equations using various methods?
• How can solutions to equations be represented?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Identities
• Multiple Representations
• Patterns/Rules
• Simplify
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Trigonometric expressions can be simplified or transformed to equivalent expressions.

• How can the graphs and attributes of trigonometric functions be used to determine certain trigonometric identities?
• What are the reciprocal, quotient, Pythagorean, cofunctions, and even/odd identities?
• What are the sum and difference identities for cosine and sine?
• Why are trigonometric expressions simplified?
• How can trigonometric identities be used to simplify trigonometric expressions?
• How can algebraic processes be applied to determine equivalent trigonometric expressions?

Equations can be used to model and solve mathematical and real-world problem situations.

• How are trigonometric equations used to model problem situations?
• What methods can be used to solve trigonometric equations?
• What are the advantages and disadvantages of various methods used to solve trigonometric equations?
• What methods can be used to justify the reasonableness of solutions?
• How can the attributes of trigonometric functions be used to determine solutions to trigonometric equations?
• How can the unit circle be used to determine solutions to trigonometric equations?
• How are the solutions to a trigonometric equation related to the period of the corresponding trigonometric function?
• How can trigonometric identities be used to solve trigonometric equations?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that identities can be verified by substituting a number in for x. However, equations that are not identities hold true for certain values of x. For example, the equation sin x = –sin x is true for x = 0 and x = , but it is not an identity.
• Some students may think that trigonometric expressions can be simplified in ways that do not use trigonometric identities, such as sin(u + v) = sin u + sin v or sin2 u = 2sin u rather than sin(u + v) = sin u cos v + cos u sin or sin 2u - 2sin u cos u.
• Some students may think that inverse trigonometric functions are the same as the reciprocalsof the trigonometric functions. For example,  is not the same as .
• Some students may incorrectly evaluate trigonometric functions or inverse trigonometric functions if they use a calculator in the wrong angle mode.

Underdeveloped Concepts:

• Students may make sign errors when using the cosine sum and difference identities.
• Students may not recognize equivalent forms of common identities, such as 1 – sin2 x = cos2 x.
• Students may not make connections between trigonometric identities and the graphs of trigonometric functions. For example, analysis of the graph of f(x) = cos x supports the trigonometric identity cos(–x) = cos x.
• Students may solve a trigonometric equation by determining a solution to the equation but may overlook other solutions over a given interval. Students may also struggle to generalize all of the solutions to the equation.

#### Unit Vocabulary

Related Vocabulary:

 Cofunction identities Common denominator Cosecant Cosine Cotangent Denominator Difference identities Double-angle identities Equation Equivalent expression Even identities Extraneous solution General solution Interval Inverse trigonometric function Numerator Odd identities Period Pythagorean identities Quotient identities Reciprocal identities Secant Simplify Sine Solve Solution Sum identities Tangent Trigonometric expression Trigonometric function Trigonometric identities Verify
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.5 Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:
P.5M Use trigonometric identities such as reciprocal, quotient, Pythagorean, cofunctions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric expressions.

Use

TRIGONOMETRIC IDENTITIES SUCH AS RECIPROCAL, QUOTIENT, PYTHAGOREAN, COFUNCTIONS, EVEN/ODD, AND SUM AND DIFFERENCE IDENTITIES FOR COSINE AND SINE

Including, but not limited to:

• Reciprocal identities
• sec x =
• csc x =
• cot x =
• Quotient identities
• tan x =
• cot x =
• Pythagorean identities
• sin2x + cos2x = 1
• tan2x + 1 = sec2x
• 1 + cot2x = csc2x
• Cofunction identities
• sin x = cos [radians]
• sin = cos(90° – ) [degrees]
• cos x = sin [radians]
• cos = sin(90° – ) [degrees]
• Even/odd properties
• The sine function is odd; therefore, sin(–x) = –sin x.
• The cosine function is even; therefore, cos(–x) = cos x.
• Sum and difference identities for cosine and sine
• sin(A + B) = sin A cos B + cos A sin B
• sin(A – B) = sin A cos B – cos A sin B
• cos(A + B) = cos A cos B – sin A sin B
• cos(A – B) = cos A cos B + sin A sin B
• Double-angle identities for cosine and sine arising from the sum and difference identities for cosine and sine
• sin2x = 2sin x cos x
• cos2x = cos2x – sin2x
• cos2x = 2cos2x – 1
• cos2x = 1 – 2sin2x

To Simplify

TRIGONOMETRIC EXPRESSIONS

Including, but not limited to:

• Using algebra skills
• Simplifying by using common factors
• Combining like terms
• Distributing
• Factoring
• Combining rational expressions by finding a common denominator
• Substituting equivalent expressions into another expression
• Writing expressions in terms of sine and cosine
• Recognizing equivalent forms of common identities
• Evaluating trigonometric functions for special values
• Verifying trigonometric identities
• Simplify only one side of the equation until it matches the other side.

Note(s):

• Geometry solved for side lengths and angle measures in triangles using the trigonometric ratios.
• Precalculus extends the use of the trigonometric functions from concrete applications involving measurement to abstract functions of real numbers.
• Algebra II performed operations (addition, subtraction, multiplication, and division) on algebraic expressions.
• Precalculus extends such symbolic manipulation to include operations on trigonometric expressions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.5N Generate and solve trigonometric equations in mathematical and real-world problems.

Generate, Solve

TRIGONOMETRIC EQUATIONS IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Mathematical problem situations
• Using algebra skills and inverse trigonometric functions
• Writing general solutions
• Finding specific solutions within a given interval
• Real-world problem situations
• Functions modeling relationships between variables
• Writing functions
• Using technology
• Graphs
• Tables
• Solving equations
• Given x, evaluate f(x).
• Given a function value, solve for x.
• Geometric situations involving triangles
• Identifying the triangle(s)
• Identifying the constants and variables
• Writing an equation
• Evaluating and solving the equation

Note(s):

• Geometry solved for side lengths and angle measures in triangles using the trigonometric ratios.
• Precalculus extends the use of the trigonometric functions from concrete applications involving measurement to abstract functions of real numbers.
• Algebra II solved exponential, logarithmic, and rational equations.
• Precalculus extends these equation-solving skills to include trigonometric equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections