# 6.2: Sine. It’s the Law.

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 5, Lesson 3.

ID: 11851

Time Required: 15 minutes

## Activity Overview

In this activity, students will explore the Law of Sines. Students will derive the formula through exploration and solve some application problems. As an extension, students will prove the Law of Sines through guided questions.

Topic: Right Triangles & Trigonometric Ratios

• Law of Sines

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Law of Sines

Students will begin this activity by looking at a triangle and investigating the ratio of the sine of an angle to the length of the opposite side. In LAW1.8xv, students are given triangle ABC\begin{align*}ABC\end{align*} with the measures of angles A,B,\begin{align*}A, B,\end{align*} and C\begin{align*}C\end{align*}, and the measure of sides a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}.

Students will collect data in the tables on their accompanying worksheet and asked what they notice about the last three columns of the table in Question 2. Discuss round off errors with the students.

Students are asked to compare the three columns to discover the Law of Sines.

## Problem 2 – Application of the Law of Sines

In Problem 2, students are asked to apply what they have learned about the Law of Sines.

## Extension – Proof of the Law of Sines

As an extension, students are asked to prove the Pythagorean theorem through guided questions.

## Solutions

1.

Sample Answers
Position a\begin{align*}a\end{align*} b\begin{align*}b\end{align*} c\begin{align*}c\end{align*} A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*}
1 5.88 8.00 5.12 47.05 93.35 39.60
2 8.97 8.00 4.01 89.99 63.44 26.57
3 8.16 8.00 3.01 81.90 76.71 21.39
4 4.84 8.00 4.00 27.41 130.24 22.35

2.

Sample Answers
Position sin(A)\begin{align*}\sin (A)\end{align*} sin(B)\begin{align*}\sin (B)\end{align*} sin(C)\begin{align*}\sin (C)\end{align*} sin(A)a\begin{align*}\frac{\sin (A)}{a}\end{align*} sin(B)b\begin{align*}\frac{\sin (B)}{b}\end{align*} sin(C)c\begin{align*}\frac{\sin (C)}{c}\end{align*}
1 0.73135 0.99829 0.63742 0.12347 0.12478 0.12449
2 1 0.89446 0.44729 0.11148 0.11180 0.11154
3 0.99002 0.97321 0.36471 0.12132 0.12165 0.12116
4 0.46035 0.76334 0.38026 0.09511 0.09541 0.09506

3. They are approximately equal (not exactly equal due to round off errors).

4. sinAa=sinBb=sinCc\begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{align*}

5. sinAa=sinBb=sinCc\begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{align*}

6. Distance from Tower A:2.45 mi\begin{align*}A: 2.45 \ mi\end{align*}; Distance from Tower B:3.99 mi\begin{align*}B: 3.99 \ mi\end{align*}

7. 36.95 ft tall

8. Distance from lighthouse A:6.65 mi\begin{align*}A: 6.65 \ mi\end{align*}; Distance from lighthouse B:9.38 mi\begin{align*}B: 9.38 \ mi\end{align*}

9. sin(A)=BDc\begin{align*}\sin (A) = \frac{BD}{c}\end{align*}

10. sin(C)=BDa\begin{align*}\sin (C)= \frac{BD}{a}\end{align*}

11. BD;csin(A)=BD\begin{align*}BD; c \cdot \sin (A) = BD\end{align*} and asin(C)=BD\begin{align*}a \cdot \sin (C) = BD\end{align*}

12. csin(A)sin(A)sin(A)a=asin(C)=asin(C)c=sin(C)c\begin{align*}c \cdot \sin (A) &= a \cdot \sin (C)\\ \sin (A) &= \frac{a \cdot \sin (C)}{c}\\ \frac{\sin (A)}{a} &= \frac{\sin (C)}{c}\end{align*}

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TI.MAT.ENG.SE.1.Trigonometry.6.2