Right Triangles and Trigonometry

Lesson 8

Math

Unit 4

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.

• 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. ​​​​​​
Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

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.

 $${\mathrm{sin}(30^\circ)}$$ $${\mathrm{sin}(45^\circ)}$$ $${\mathrm{sin}(60^\circ)}$$ $${\mathrm{cos}(30^\circ)}$$ $${\mathrm{cos}(45^\circ)}$$ $${\mathrm{cos}(60^\circ)}$$

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.

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.

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..

Lesson 7

Lesson 9

Lesson Map

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