Right Triangles and Trigonometry

Lesson 8

Math

Unit 4

10th Grade

Lesson 8 of 19

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

  • G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.

Foundational Standards

  • G.CO.C.10

Criteria for Success


  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. ​​​​​​
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Anchor Problems


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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Target Task


FInd the value of $${\theta}$$ that makes each statement true.

$$\mathrm{sin}{\theta}=\mathrm{cos}32$$

$$\mathrm{cos}{\theta}=\mathrm{sin}({\theta}+20)$$

References

EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 27Exit Ticket, Question #1

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..

Additional Practice


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

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Lesson 7

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Lesson 9

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Right Triangle Properties and Side-Length Relationships

Topic B: Right Triangle Trigonometry

Topic C: Applications of Right Triangle Trigonometry

Topic D: The Unit Circle

Topic E: Trigonometric Ratios in Non-Right Triangles

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