Unit Circle and Trigonometric Functions

Lesson 8

Objective

Evaluate transformations of sine, cosine, and tangent such as $${2{\pi-x}}$$$${\pi-x}$$, and $${\pi+x}$$.

Common Core Standards

Core Standards

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  • F.TF.A.1 — Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

  • F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.

Foundational Standards

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  • G.C.B.5

Criteria for Success

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  1. Visualize transformations of sine, cosine, and tangent on the unit circle.
  2. Use transformations of sine, cosine, and tangent to more quickly evaluate particular angles. 

Anchor Problems

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Problem 1

FInd the value of the expression below:

$${\mathrm{sin}\frac{11\pi}{6}}$$

Guiding Questions

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References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1Lesson 1, Exercise 2

Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 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..

Problem 2

Explain why each of the following expressions are equivalent:

$${\mathrm{sin}\theta=\mathrm{sin}(x-\theta)}$$

$${\mathrm{cos}\theta=\mathrm{cos}(2\pi-\theta)}$$

$${\mathrm{cos}\theta=\mathrm{cos}(-\theta)}$$

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Include problems evaluating angles close to $${\pi}$$ and $$2{\pi}$$ and ask students to visualize them multiple ways.
  • Include problems asking about the equivalence of different expressions, as in the second anchor problem.
  • Include problems evaluating expressions written multiple ways and explaining why they are the same, such as $$\mathrm{cos}\frac{3{\pi}}{4}$$ and $$\mathrm{cos}\frac{5{\pi}}{4}$$.

Target Task

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Problem 1

Evaluate the following trigonometric expressions and explain how you used the unit circle to determine your answer.

a.   $${\mathrm{sin}\left(\pi+\frac{\pi}{3}\right)}$$

b.   $${\mathrm{cos}\left(2\pi-\frac{\pi}{6}\right)}$$

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1Exit Ticket, Question #1

Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 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..

Problem 2

Corinne says that for any real number $${\theta}$$, $$\mathrm{cos}{\theta}=\mathrm{cos}{\theta}-\pi$$. Is she correct? Explain how you know.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1Exit Ticket, Question #2

Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 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..