Right Triangles and Trigonometry
Lesson 8
Objective
Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°.
Common Core Standards
Core Standards
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G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.
Foundational Standards
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G.CO.C.10
Criteria for Success
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- Describe that the value of sine approaches 1 and the value of the cosine approaches 0 as an angle measure approaches 90°.
- Describe that the value of sine approaches 0 and the value of the cosine approaches 1 as an angle measure approaches 0°.
- Derive the relationship between sine and cosine of complementary angles in right triangles using the reference angles of 30°/60°, 45°/45°, 90°/0°.
- Extend the relationship of sine and cosine of complementary angles to non-reference angles in right triangles.
Anchor Problems
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Problem 1
Below are three right triangles. Assume the value of the hypotenuse of each triangle is $$1$$.
a) Using what you know about special right triangles, find the length of each side.
b) Fill in the chart describing the sine and cosine of each measure below.
| $${\mathrm{sin}(30^\circ)}$$ | $${\mathrm{sin}(45^\circ)}$$ | $${\mathrm{sin}(60^\circ)}$$ |
| $${\mathrm{cos}(30^\circ)}$$ | $${\mathrm{cos}(45^\circ)}$$ | $${\mathrm{cos}(60^\circ)}$$ |
Guiding Questions
Problem 2
Using the cosine and sine values from the table in Anchor Problem #1, identify trigonometric ratios that are the same. Then, write a conjecture about how the sine is related to the cosine of complementary angles.
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.
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Illustrative Mathematics Sine and Cosine of Complementary Angles
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EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 27
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CK-12 Sine and Cosine of Complementary Angles
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Only use questions about sine and cosine
Target Task
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FInd the value of $${\theta}$$ that makes each statement true.
$$\mathrm{sin}{\theta}=\mathrm{cos}32$$
$$\mathrm{cos}{\theta}=\mathrm{sin}({\theta}+20)$$
References
Geometry > Module 2 > Topic E > Lesson 27 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..