Curriculum  /  Math  /  11th Grade  /  Unit 7: Trigonometric Identities and Equations  /  Lesson 2

Trigonometric Identities and Equations

Lesson 2

Math

Unit 7

11th Grade

Lesson 2 of 16

Objective

Derive and use the Pythagorean identity to write equivalent expressions.

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

  • 8.G.B.6
  • 8.G.B.7
  • 8.G.B.8
  • G.SRT.C.8

Criteria for Success

  • Derive the Pythagorean identity of $${{\mathrm{sin}^{2}x+\mathrm{cos}^2x=1}}$$ by connecting sines and cosines on the unit circle with right triangles.
  • Use appropriate notation to write powers of trigonometric functions.
  • Derive the remaining Pythagorean identities through manipulation of $${{\mathrm{sin}^{2}x+\mathrm{cos}^2x=1}}$$.
  • Identify any domain restrictions that follow from the manipulation of Pythagorean identities. Describe why these domain restrictions exist. 
  • Use Pythagorean identities to evaluate trigonometric functions.

Tips for Teachers

  • Students will likely struggle with the conventions of how to write the square of trigonometric ratios, and there are a bunch of common errors that can result from this misunderstanding. Require that students use precise mathematical notation so as not to get confused. Here is a great Dr. Math article from The Math Forum on the topic: “Interpreting Expressions Involving Cosine Squared.”
  • Quick Questions on Proving Trig Identities by Sam Shah is an article on proving trig identities that is useful to read before going through this with students.
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Anchor Problems

25-30 minutes

Problem 1

Below is a unit circle and a generic triangle with the radius as hypotenuse.

Write the Pythagorean Theorem in terms of sine and cosine. 

Note: The square of cosine is written as “$${\mathrm{cos}^2x}$$.”

Guiding Questions

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Problem 2

Part A: Solve the Pythagorean identity for $${\mathrm{sin}^2\theta}$$ and for $${\mathrm{cos}^2\theta}$$. 
Part B: Solve the Pythagorean identity for $${\mathrm{tan}^2\theta}$$. 

Guiding Questions

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Problem 3

Suppose that $${cos{\theta}={2\over5}}$$ and that $${\theta}$$ is in the 4th quadrant. Find $$\mathrm{sin}{\theta}$$ and $$\mathrm{tan}{\theta}$$ exactly.

Guiding Questions

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References

Illustrative Mathematics Finding Trig Values

Finding Trig Values, accessed on May 21, 2018, 1:39 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

5-10 minutes

April claims that $${1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}={1\over{\mathrm{sin}^2({{\theta}})}}}$$ is an identity for all real numbers $${{\theta}}$$ that follows from the Pythagorean identity.

  1. For which values of $${{\theta}}$$ are the two functions $$f({{\theta}})=1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}$$ and $$g({{\theta}})={1\over{\mathrm{sin}^2({{\theta}})}}$$ defined?
  2. Show that the equation $${1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}={1\over{\mathrm{sin}^2({{\theta}})}}}$$ follows from the Pythagorean identity.

References

EngageNY Mathematics Algebra II > Module 2 > Topic B > Lesson 15 — Exit Ticket, Questions a and b

Algebra II > Module 2 > Topic B > Lesson 15 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..

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 verifying identities using the Pythagorean identities.
  • Include problems identifying domain restrictions for different types of identities.
  • Include problems that do not have a linear final term, such as $${\mathrm{cos}^2(x)-1=-\mathrm{sin}^2x}$$.
  • Include error analysis problems where two expressions are not equivalent.

Next

Verify trigonometric identities using Pythagorean and reciprocal identities.

Lesson 3
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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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