Trigonometric Identities and Equations

Lesson 10

Math

Unit 7

11th Grade

Lesson 10 of 16

Objective


Solve trigonometric equations using identities.

Common Core Standards


Core Standards

  • F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
  • 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.3
  • F.TF.A.4

Criteria for Success


  1. Solve trigonometric equations using properties of the unit circle and graphs of functions.
  2. Solve trigonometric equations using Pythagorean identities. 

Tips for Teachers


This lesson intentionally avoids excessively complicated algebra or convoluted reasoning. Students should focus on reasoning about the unit circle and applying Pythagorean identities. While there are plenty of more complex ways to apply the idea of solving trig equations, students will be better served by focusing on a deeper dive into a few essential skills.

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Anchor Problems


Problem 1

What is the general solution for the equation $${\mathrm{sin}x=\mathrm{csc}x}$$?

Find the solution algebraically and graphically.

Guiding Questions

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Problem 2

What is the general solution for $${\mathrm{sin}^2x+\mathrm{cos}x+1=0}$$?

Guiding Questions

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Target Task


What is the general solution for these equations?

$${-\mathrm{sin}x=\mathrm{cos}x}$$

$${\mathrm{sin}^2x+2\mathrm{cos}x-2=0}$$

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 that ask students to interpret functions graphically.
  • Include problems involving reciprocal and Pythagorean identities.
  • Include problems that students can visualize using the unit circle, for instance solving $${\mathrm{sin}x=\mathrm{cos}x}$$ by thinking about where sine and cosine are equivalent on the unit circle.
  • Here are some resources to find problems: 
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Lesson 9

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Lesson 11

Lesson Map

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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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