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# Unit Circle and Trigonometric Functions

## 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.

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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:

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

#### 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:

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

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

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

## 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 $\mathrm{cos}\frac{5{\pi}}{4}$.

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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..