Lesson 1 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

Derive and verify trigonometric identities using transformations and equivalence of functions.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Trigonometric Functions

F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Foundational Standards

The foundational standards covered in this lesson

F.TF.A.2

Trigonometric Functions

F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
F.TF.A.3

Trigonometric Functions

F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
F.TF.A.4

Trigonometric Functions

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define an identity as an equation that is true for all values within a certain domain.
Use rigid transformations to write cofunction identities.
Use rigid transformations to write even and odd identities.
Use reciprocal and quotient identities to prove equivalence of expressions using trigonometric functions.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

A good initial first step is to give students the associative property or the commutative property in an algebraic example and describe why this is an identity.
Students will not need to memorize all of the identities shown in this unit, but they will need to memorize the reciprocal identities. The most common identities will be used often enough that students will likely have them memorized by the end of the unit.
As students are verifying an identity, they should only work the left-hand side (LHS) or the right-hand side (RHS) to show that one is equal to the other.
Sam Shah has some great ideas and materials on verifying trig identities in his post, Dan Meyer Says Jump and I Shout How High?

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Below is a graph of a sine function and a cosine function.

Which of the following equations are true statements?

$${\mathrm{sin}(-\theta)=-\mathrm{sin}(\theta)}$$

$${\mathrm{sin}\left({\pi\over2}-\theta\right)=\mathrm{cos}(\theta)}$$

$${\mathrm{sin}(2\pi-\theta)=\mathrm{sin}(\theta)}$$

Guiding Questions

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Problem 2

Verify the identity using definitions of functions.

$${{\mathrm{cot}(x)\over{\mathrm{csc}(x)}}=\mathrm{cos}(x)}$$

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Verify the identity.

$${{\mathrm{tan}(x)\over{\mathrm{sin}(x)}}=\mathrm{sec}(x)}$$

On what domain does this equation hold?

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems using reciprocal identities to verify other identities, modeled in Anchor Problem #2
Include problems asking students to verify if two graphs are equivalent and on what domain this is true.
Include problems that use the unit circle to verify identities using sines and cosines.

Illustrative Mathematics Trigonometric Identities and Rigid Motions
Illustrative Mathematics Properties of Trigonometric Functions
EngageNY Mathematics Algebra II > Module 2 > Topic B > Lesson 14 — Problem Set and Exercises (Use these to evaluate using the identities in Anchor Problem #1)

Lesson 2

Lesson Map

Topic A: Basic Trigonometric Identities and Equivalent Expressions

Derive and verify trigonometric identities using transformations and equivalence of functions.

F.TF.C.8

Derive and use the Pythagorean identity to write equivalent expressions.

F.TF.C.8

Verify trigonometric identities using Pythagorean and reciprocal identities.

F.TF.C.8

Topic B: Solve Trigonometric Equations

Find angle measures using inverse trig functions in right triangles.

F.TF.B.6 F.TF.B.7

Analyze inverse trigonometric functions graphically.

F.BF.B.4.D F.IF.C.7.E F.TF.B.6 F.TF.B.7

Solve linear trigonometric equations.

F.TF.B.7

Solve linear trigonometric equations using $$u$$-substitution.

F.TF.B.7

Use inverse trigonometric functions to solve contextual problems.

F.TF.B.7

Solve quadratic trigonometric equations.

F.TF.B.6 F.TF.B.7

Solve trigonometric equations using identities.

F.TF.B.7 F.TF.C.8

Topic C: Advanced Identities and Solving Trigonometric Equations

Evaluate expressions using sum and difference formulas.

F.TF.C.9

Solve equations and prove identities using sum and difference formulas.

F.TF.C.9

Derive double angle formulas and use them to solve equations and prove identities.

F.TF.C.9

Use trigonometric identities to analyze graphs of functions.

F.TF.C.8 F.TF.C.9

Topic D: Applications and Extensions of Trigonometric Functions

Use the Law of Sines to find missing side lengths and angle measures in acute triangles.

G.SRT.D.10

Find missing side lengths and angle measures using the Law of Cosines in acute triangles.

G.SRT.D.10

Trigonometric Identities and Equations

Objective

Common Core Standards

Core Standards

Trigonometric Functions

Foundational Standards

Trigonometric Functions

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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