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

Trigonometric Identities and Equations

Lesson 11

Math

Unit 7

11th Grade

Lesson 11 of 16

Objective

Evaluate expressions using sum and difference formulas.

Common Core Standards

Core Standards

  • F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Foundational Standards

  • F.TF.A.3

Criteria for Success

  1. Identify why sum and difference formulas are necessary.
  2. Find the sine, cosine, and tangent of angles using the sum and difference formulas. 
  3. Solve problems using sum and difference formulas.

Tips for Teachers

There is a geometric proof of the sum and difference formulas. Here is the Kahn Academy version, Proof of the sine angle addition identity.

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Anchor Problems

25-30 minutes

Problem 1

Are these expressions equivalent? Why or why not?

a.  $${\mathrm{sin}(a+b)}$$          b.  $${\mathrm{sin}a+\mathrm{sin}b}$$

How can you justify your reasoning when $${a={\pi\over6}}$$ and $${b={\pi\over3}}$$? Use the unit circle shown below to justify your thinking.

Guiding Questions

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Problem 2

Below are the sum and difference formulas for sine, cosine, and tangent.

$${\mathrm{sin}(a+b)=\mathrm{sin}a\mathrm{cos}b+\mathrm{cos}a\mathrm{sin}b}$$

$${\mathrm{sin}(a-b)=\mathrm{sin}a\mathrm{cos}b-\mathrm{cos}a\mathrm{sin}b}$$

 

$${\mathrm{cos}(a+b)=\mathrm{cos}a\mathrm{cos}b-\mathrm{sin}a\mathrm{sin}b}$$

$${\mathrm{cos}(a-b)=\mathrm{cos}a\mathrm{cos}b+\mathrm{sin}a\mathrm{sin}b}$$

 

$${\mathrm{tan}(a+b)={\mathrm{tan}a+\mathrm{tan}b\over{1-\mathrm{tan}a\mathrm{tan}b}}}$$

$${\mathrm{tan}(a-b)={\mathrm{tan}a-\mathrm{tan}b\over{1+\mathrm{tan}a\mathrm{tan}b}}}$$

 

Use both sum and difference formulas to find the value of:

$${\mathrm{sin}(75^{\circ})}$$

$${\mathrm{tan}(15^{\circ})}$$

$${\mathrm{cos}(105^{\circ})}$$

Guiding Questions

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Problem 3

Find $${\mathrm{cos}(a+b)}$$ given that $${\mathrm{cos}(a)=-{4\over5}}$$ with $${\pi \leq a \leq {3\pi\over2}}$$ and $${\mathrm{sin}(b)={5\over13}}$$ with $${0 \leq b \leq {\pi\over2}}$$.

Guiding Questions

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Target Task

5-10 minutes

Problem 1

Find $${\mathrm{sin}(u-v)}$$ given $${\mathrm{sin}(u)={8\over17}}$$ with $${0\leq u \leq {\pi\over2}}$$ and $${\mathrm{cos}(v)=-{24\over25}}$$ with $${\pi \leq v \leq {3\pi\over2}}$$.

Problem 2

Evaluate: 

$${\mathrm{sin}(-15^{\circ})}$$

$${\mathrm{cos}(15^{\circ})}$$

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.

  • Give students the sum formulas and the understanding that $${a-b=a+(-b)}$$. Ask students to verify the difference formulas using this understanding.
  • Evaluate sine, cosine, and tangent using sum and difference formulas.
  • Describe when sum and difference formulas are not appropriate to use. Use problems on pages 71-77 of Trigonometry by Michael Corral.

Next

Solve equations and prove identities using sum and difference formulas.

Lesson 12
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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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