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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Problem Set


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


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..

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Lesson 7

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Lesson 9

Lesson Map

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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