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6.2: Sine. It’s the Law.

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}Y=\end{align*}, selecting Open…, and selecting the file. You are given \begin{align*}\triangle ABC\end{align*} with the measure of all angles and sides calculated.

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

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

Position \begin{align*}\sin(A)\end{align*} \begin{align*}\sin(B)\end{align*} \begin{align*}\sin(C)\end{align*} \begin{align*}\frac{\sin(A)}{a}\end{align*} \begin{align*}\frac{\sin(B)}{b}\end{align*} \begin{align*}\frac{\sin(C)}{c}\end{align*}
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3. What do you notice about the last three columns of the table in Question 2?

4. Make a conjecture relating \begin{align*}\frac{\sin A}{a}\end{align*}, \begin{align*}\frac{\sin B}{b}\end{align*}, and \begin{align*}\frac{\sin C}{c}\end{align*} .

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 \begin{align*}A\end{align*} spots a fire \begin{align*}52^\circ\end{align*} SE and the observer in tower \begin{align*}B\end{align*} spots the same fire \begin{align*}29^\circ\end{align*} SW. Find the distance of the fire from each tower.

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

8. A boat is spotted by lighthouse \begin{align*}A\end{align*} at \begin{align*}25^\circ\end{align*} NE and spotted by lighthouse \begin{align*}B\end{align*} at \begin{align*}50^\circ\end{align*} 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 \begin{align*}\frac{\sin(A)}{a}=\frac{\sin(C)}{c}\end{align*}. You can use similar methods to show that \begin{align*}\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\end{align*} and \begin{align*}\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\end{align*} . You are given \begin{align*}\triangle ABC\end{align*}, altitude \begin{align*}BD\end{align*}, and sides \begin{align*}a\end{align*} and \begin{align*}c\end{align*}.

9. Using right triangular trigonometry, what is the sine ratio for \begin{align*}\angle A\end{align*}?

10. Using right triangular trigonometry, what is the sine ratio for \begin{align*}\angle C\end{align*}?

11. What side is common to the sine of \begin{align*}A\end{align*} and the sine of \begin{align*}C\end{align*}? Solve for this common side in the ratio for sine of \begin{align*}A\end{align*} and sine of \begin{align*}C\end{align*}.

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 \begin{align*}\frac{\sin (A)}{a}=\frac{\sin (C)}{c}\end{align*}.

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Date Created:
Feb 23, 2012
Last Modified:
Nov 04, 2014
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