Curriculum  /  Math  /  11th Grade  /  Unit 7: Trigonometric Identities and Equations  /  Lesson 3

Trigonometric Identities and Equations

Lesson 3

Math

Unit 7

11th Grade

Lesson 3 of 16

Objective

Verify trigonometric identities using Pythagorean and reciprocal identities.

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

  • A.APR.D.6
  • A.REI.A.1
  • A.SSE.A.2
  • A.SSE.B.3.A

Criteria for Success

  1. Recognize Pythagorean identities to rewrite expressions.
  2. Factor simple trigonometric expressions using the GCF.
  3. Find common denominators of trigonometric expressions.
  4. Multiply by conjugates to rewrite trigonometric expressions.
  5. Recognize appropriate strategies to rewrite trigonometric expressions.
  6. Prove trigonometric identities by applying appropriate strategies to rewrite expressions.
  7. Identify domain restrictions of identities.

Tips for Teachers

  • In the guiding questions, the idea of substituting one variable for a more complicated function is introduced. Determine whether this is appropriate for your class. $$u$$-substitution will come up later on in the unit. Solving Trig Equations Using U-Substitution on Youtube is a great introduction to the idea with a relatively simple sine equation.
  • As students are verifying an identity, they should only work the left-hand side or the right-hand side to show that one is equal to the other.
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Anchor Problems

25-30 minutes

Problem 1

Rewrite the expression so that it has only one term.

$${\mathrm{cos}x-\mathrm{cos}x\mathrm{sin}^2x}$$

Guiding Questions

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Problem 2

Amy is trying to rewrite the expression $${\mathrm{sin}x+\mathrm{cos}x{\mathrm{cot}x}}$$ so that it is easier to graph. 

She first recognized that $${\mathrm{cot}x}$$ can be rewritten as $${\mathrm{cos}x\over{\mathrm{sin}x}}$$. 

Rewrite the expression, applying what Amy noticed, then write the expression as one term. 

Guiding Questions

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Problem 3

Prove the following is an identity:

$${{{\mathrm{sin}^2x}\over{1-\mathrm{cos}x}}=1+\mathrm{cos}x}$$

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 2 > Topic B > Lesson 16 — Opening Exercise

Algebra II > Module 2 > Topic B > Lesson 16 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Target Task

5-10 minutes

Problem 1

Prove and identify the domain of the following identity.

$${\mathrm{tan}\theta\mathrm{csc}\theta=\mathrm{sec}\theta}$$

Problem 2

Rewrite the following trigonometric expression so that it has only one term.

$${\mathrm{sec}x\mathrm{cot}x-\mathrm{cot}x\mathrm{cos}x}$$

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 determining whether two graphs are equivalent and noting any domain on which either function is undefined.
  • Include non-examples of identities for students to distinguish from true identities.

Next

Find angle measures using inverse trig functions in right triangles.

Lesson 4
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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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