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# 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.

## Problem 1 – Law of Sines

Open the Cabri Jr. application by pressing APPS and selecting CabriJr. Open the file LAW1 by pressing $Y=$, selecting Open…, and selecting the file. You are given $\triangle ABC$ with the measure of all angles and sides calculated.

1. Grab and drag point $B$ (use the ALPHA button to grab the point), and record the values of $a, \ b, \ c, \ \angle A, \angle B$, and $\angle C$. Repeat this three more times.

Position $a$ $b$ $c$ $A$ $B$ $C$
1
2
3
4

2. On the calculator home screen calculate $\sin(A)$, $\sin(B)$, and $\sin(C)$. Then, calculate the following ratios: $\frac{\sin(A)}{a}$, $\frac{\sin(B)}{b}$, and $\frac{\sin(C)}{c}$.

Position $\sin(A)$ $\sin(B)$ $\sin(C)$ $\frac{\sin(A)}{a}$ $\frac{\sin(B)}{b}$ $\frac{\sin(C)}{c}$
1
2
3
4

3. What do you notice about the last three columns of the table in Question 2?

4. Make a conjecture relating $\frac{\sin A}{a}$, $\frac{\sin B}{b}$, and $\frac{\sin C}{c}$ .

## Problem 2 – Application of the Law of Sines

5. State the Law of Sines.

6. The distance between two fire towers is 5 miles. The observer in tower $A$ spots a fire $52^\circ$ SE and the observer in tower $B$ spots the same fire $29^\circ$ SW. Find the distance of the fire from each tower.

7. A tree leans $20^\circ$ from vertical and at a point 50 ft. from the tree the angle of elevation to the top of the tree it $29^\circ$. Find the height, $h$, of the tree.

8. A boat is spotted by lighthouse $A$ at $25^\circ$ NE and spotted by lighthouse $B$ at $50^\circ$ NW. The lighthouses are 10 miles apart. What is the distance from the boat to each lighthouse?

## Extension – Proof of the Law of Sines

We will now prove the Law of Sines. We will prove that $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$. You can use similar methods to show that $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}$ and $\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$ . You are given $\triangle ABC$, altitude $BD$, and sides $a$ and $c$.

9. Using right triangular trigonometry, what is the sine ratio for $\angle A$?

10. Using right triangular trigonometry, what is the sine ratio for $\angle C$?

11. What side is common to the sine of $A$ and the sine of $C$? Solve for this common side in the ratio for sine of $A$ and sine of $C$.

12. Since the side from Exercise 13 is common to both equations we can set them equal to each other. Set your two equations equal and try to show that $\frac{\sin (A)}{a}=\frac{\sin (C)}{c}$.

Feb 23, 2012

Nov 04, 2014