# Right Triangles and Trigonometry

Lesson 16

Math

Unit 4

Lesson 16 of 19

## Objective

Derive the area formula for any triangle in terms of sine.

## Common Core Standards

### Core Standards

• G.SRT.D.9 — Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

• 6.G.A.1

## Criteria for Success

1. Use the area formula for triangles to determine missing information in non-right triangles.
2. Identify that an auxiliary altitude is necessary in non-right triangles to represent the height.
3. Write the measurement of the height of a non-right triangle in terms of sine of the angle opposite to the height.
4. Assign the variables $A$, $B$, and $C$ to each of the vertices of a non-right triangle.
5. Assign the variables $a$$b$, and $c$ to each side opposite vertices $A$, $B$, and $C$ respectively in non-right triangles.
6. Generalize the area formula for non-right triangles as $\mathrm{Area}=\frac{1}{2}ab\mathrm{sin}(C)$.
7. Verify that the formula works for any non-right triangle once $a$$b$, and $c$  sides and $A$, $B$, and $C$ angles are defined.

## Tips for Teachers

This post on Math Central is helpful in developing knowledge on the area formula in terms of sine for triangles.

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## Anchor Problems

### Problem 1

Find the area of the following triangle. ### Problem 2

Write a formula for the area of this non-right triangle. ## 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.

• Include problems where the students are given the area and need to find missing information about the non-right triangle.
• Include problems where the students are given the area and need to find missing information about the non-right triangle.

Explain why ${\frac{1}{2}ab\mathrm{sin}(C)}$ gives the area of a triangle with sides $a$ and $b$ and included angle $C$.

#### References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic B > Lesson 7Exit Ticket, Question #2

Precalculus and Advanced Topics > Module 4 > Topic B > Lesson 7 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 Fishtank Learning, Inc.

Lesson 15

Lesson 17

## 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