Right Triangles and Trigonometry

Lesson 17

Objective

Verify algebraically and find missing measures using the Law of Sines.

Common Core Standards

Core Standards

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  • G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.

Foundational Standards

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  • 7.RP.A.2

Criteria for Success

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  1. Describe why trigonometric ratios cannot be applied directly in non-right triangles.
  2. Describe how auxiliary altitudes are beneficial in establishing additional relationships in non-right triangles. 
  3. Write the height of the altitude in terms of sine of angles in the non-right triangle. 
  4. Using the transitive property, establish that the Law of Sines of $${\frac{\mathrm{sin}A}{a}=\frac{\mathrm{sin}b}{b}=\frac{\mathrm{sin}c}{c}}$$ describes the relationships in non-right triangles. 
  5. Use this relationship to solve triangles.
  6. Describe the ambiguous case for the Law of Sines.

Tips for Teachers

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Anchor Problem #2 will require the teacher to do a significant amount of modeling to define and algebraically verify the Law of Sines. Reference EngageNY, Geometry, Module 2, Lesson 32: Teacher Version (Discussion) for more guidance on this. 

Anchor Problems

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Problem 1

a. Find the lengths of $$d$$ and $$e$$.

b. Why can't you use the same method to find the lengths of $$x$$ and $$y$$?

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 32Opening Exercise

Geometry > Module 2 > Topic E > Lesson 32 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..

Modified by The Match Foundation, Inc.

Problem 2

There is a rule, the Law of Sines, which allows you to use trigonometric ratios to determine side lengths and angles of non-right triangles. 

Law of Sines: 
Given $${\triangle ABC}$$,

Using the two congruent triangles below and the altitudes drawn, show that the Law of Sines is true.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 32Discussion

Geometry > Module 2 > Topic E > Lesson 32 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..

Problem 3

In $${\triangle ABC}$$$${a=9}$$$${c=12}$$, and $${\angle A=30^\circ}$$. Determine the missing angle and side measurements.

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.

  • Include problems where students reenact the Law of Sines on their own. 
  • Be sure to include an example that you highlight during independent practice that deals with the ambiguous case where a unique triangle is not given via the Law of Sines.

Target Task

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Use the Law of Sines to find the lengths $$b$$ and $$c$$ in the triangle below. Round answers to the nearest tenth as necessary.

References