Trigonometric Identities and Equations

Lesson 14

Math

Unit 7

11th Grade

Lesson 14 of 16

Objective


Use trigonometric identities to analyze graphs 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.
  • 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
  • F.TF.A.4

Criteria for Success


  1. Reason about the solutions to trigonometric equations without solving for exact solutions.
  2. Rewrite equations to analyze graphs of functions.
  3. Identify intersection points of graphs from functions.
  4. Reason about domain restrictions to find intersection points of graphs. 

Tips for Teachers


This lesson synthesizes skills from the unit as a whole. While many students will have already mastered these skills, this provides an opportunity to explore more deeply as well as space to review skills and target problems toward what students have struggled with.

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Anchor Problems


Problem 1

Given that

$${f(x)={{\mathrm{cos}x+\mathrm{sin}x}\over\mathrm{cos}(2x)}}$$

$${g(x)={1\over{\mathrm{cos}x\mathrm{sin}x}}}$$

Answer the following questions:

  1. Identify where both functions are undefines over the domain of $${0 \leq x \leq 2\pi}$$.
  2. Show that, where defined, $${f(x)= g(x)}$$.

Guiding Questions

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Problem 2

Fill in the blanks below using a number from $$1$$ to $$9$$ (no repeating), to find the equation whose solution is the largest value of $$x$$ (from $$0$$ to $$360$$, or $$0$$ to $${2\pi}$$). What numbers will give you the smallest value of $$x$$?

Guiding Questions

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References

Open Middle Solving Trigonometric Equations

Solving Trigonometric Equations by is made available on Open Middle under the CC BY-NC-SA 4.0 license. Accessed May 21, 2018, 2:59 p.m..

Target Task


Given:

$${{f(x)}=\mathrm{cos}(2x)}$$

$${g(x)=-\mathrm{cos}(x)}$$

a.   Describe the period, amplitude, and midline of the function $${f(x)}$$.

b.   Where is $${f(x)}=g(x)$$ over the domain of $${0\leq x \leq 2\pi}$$?

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 finding where two functions are equivalent.
  • Include graphing problems that require rewriting using identities.
  • Include problems asking students to compare values reasoning with the unit circle without actually evaluating, like Anchor Problem #2.
  • Include problems reviewing skills that students have struggled with in the unit as a whole.
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Lesson 13

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Lesson 15

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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