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.
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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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- Visualize transformations of sine, cosine, and tangent on the unit circle.
- 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
References
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
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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EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1
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Problem Set
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
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
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..