Curriculum  /  Math  /  11th Grade  /  Unit 7: Trigonometric Identities and Equations  /  Lesson 1

Trigonometric Identities and Equations

Lesson 1

Math

Unit 7

11th Grade

Lesson 1 of 16

Objective

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

Common Core Standards

Core Standards

  • 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

  • F.TF.A.2
  • F.TF.A.3
  • F.TF.A.4

Criteria for Success

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

Tips for Teachers

  • 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

25-30 minutes

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

5-10 minutes

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. 

Next

Derive and use the Pythagorean identity to write equivalent expressions.

Lesson 2
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Basic Trigonometric Identities and Equivalent Expressions

Topic B: Solve Trigonometric Equations

Topic C: Advanced Identities and Solving Trigonometric Equations

Topic D: Applications and Extensions of Trigonometric Functions

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