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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  1. Describe that the value of sine approaches 1 and the value of the cosine approaches 0 as an angle measure approaches 90°. 
  2. Describe that the value of sine approaches 0 and the value of the cosine approaches 1 as an angle measure approaches 0°. 
  3. Derive the relationship between sine and cosine of complementary angles in right triangles using the reference angles of 30°/60°, 45°/45°, 90°/0°. 
  4. 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

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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

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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.

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