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# 9.3: Ratios of Right Triangles

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 8, Lesson 5.

## Problem 1 – Exploring Right Triangle Trigonometry

We will begin this activity by looking at the definitions of the sine, cosine, and tangent of a right triangle. Start the Learning Check application by pressing APPS and selecting LearnChk. Open the file Right Triangle Trigonometry. You are given the definition for the sine, cosine, and tangent of a right triangle. Copy the definitions onto your worksheet.

1. What is the definition of sinA\begin{align*}\sin A\end{align*} for right ABC\begin{align*}\triangle{ABC}\end{align*}?

2. What is the definition of cosA\begin{align*}\cos A\end{align*} for right ABC\begin{align*}\triangle{ABC}\end{align*}?

3. What is the definition of tanA\begin{align*}\tan A\end{align*} for right ABC\begin{align*}\triangle{ABC}\end{align*}?

Answer the following questions about sine, cosine, and tangent for ABC\begin{align*}\triangle{ABC}\end{align*}.

4. What is sinA\begin{align*}\sin A\end{align*}?

5. What is cosA\begin{align*}\cos A\end{align*}?

6. What is tanA\begin{align*}\tan A\end{align*}?

7. What is sinB\begin{align*}\sin B\end{align*}?

8. What is cosB\begin{align*}\cos B\end{align*}?

9. What is tanB\begin{align*}\tan B\end{align*}?

## Problem 2 – Exploring the Sine Ratio of a Right Triangle

For this problem, we will investigate the sine ratio. Start the Cabri Jr. application by pressing A\begin{align*}A\end{align*} and selecting Cabri Jr. Open the file TRIG by pressing Y=\begin{align*}Y=\end{align*}, selecting Open..., and selecting the file. You are given right triangle ABC\begin{align*}ABC\end{align*}.

10. Grab and drag point B\begin{align*}B\end{align*}. Record the data you collected in the table on the next page. Leave the last column blank for now.

Position BC\begin{align*}BC\end{align*} AB\begin{align*}AB\end{align*} BCAB\begin{align*}\frac{BC}{AB}\end{align*} sin1BCAB\begin{align*}\sin^{-1}\frac{BC}{AB}\end{align*}
1
2
3
4

11. What do you notice about the ratio of BC\begin{align*}BC\end{align*} to AB\begin{align*}AB\end{align*}?

12. Did A\begin{align*}\angle{A}\end{align*} change when you moved point B\begin{align*}B\end{align*} in ABC\begin{align*}\triangle{ABC}\end{align*}?

Because the ratio remains the same and A\begin{align*}\angle{A}\end{align*} remains fixed, we can use the ratio of BC\begin{align*}BC\end{align*} to AB\begin{align*}AB\end{align*} to find the measurement of A\begin{align*}\angle{A}\end{align*}. To do this, we will use the definition of sine and the inverse of sine. By definition, sinA=BCAB\begin{align*}\sin A = \frac{BC}{AB}\end{align*}. To find the measurement of A\begin{align*}\angle{A}\end{align*}, we use the inverse of sine to get the formula A=sin1BCAB\begin{align*}A = \sin^{-1}\frac{BC}{AB}\end{align*}. Exit Cabri Jr. and go to the home screen to find the inverse sine of BCAB\begin{align*}\frac{BC}{AB}\end{align*}. Record this into the last column of the table above.

13. What is the measurement of A\begin{align*}\angle{A}\end{align*}?

14. What is the measurement of B\begin{align*}\angle{B}\end{align*}?

## Problem 3 – Exploring the Cosine Ratio of a Right Triangle

For this problem, we will investigate the sine ratio. Start the Cabri Jr. application and open the file TRIG. You are given right triangle ABC\begin{align*}ABC\end{align*}.

15. Collect data for four positions of point B\begin{align*}B\end{align*} like that which was done in Problem 2.

Position AC\begin{align*}AC\end{align*} AB\begin{align*}AB\end{align*} ACAB\begin{align*}\frac{AC}{AB}\end{align*} cos1ACAB\begin{align*}\cos^{-1}\frac{AC}{AB}\end{align*}
1
2
3
4

Because the ratio remains the same, and A\begin{align*}\angle{A}\end{align*} remains fixed, we can use the ratio of AC\begin{align*}AC\end{align*} to AB\begin{align*}AB\end{align*} to find the measurement of A\begin{align*}\angle{A}\end{align*}. To do this, we will use the definition of cosine and the inverse of cosine. By definition, cosA=ACAB\begin{align*}\cos A = \frac{AC}{AB}\end{align*}. To find the measurement of A\begin{align*}\angle{A}\end{align*}, we use the inverse of cosine to get the formula A=cos1ACAB\begin{align*}A = \cos^{-1}\frac{AC}{AB}\end{align*}. Exit Cabri Jr. and go to the home screen to find the inverse cosine of ACAB\begin{align*}\frac{AC}{AB}\end{align*}. Record this into the last column of the table above.

16. What is the measurement of A\begin{align*}\angle{A}\end{align*}?

17. What is the measurement of B\begin{align*}\angle{B}\end{align*}?

18. How would you solve an equation of the form tanA=BCAC\begin{align*}\tan A = \frac{BC}{AC}\end{align*}?

## Problem 4 – Applying the Sine, Cosine, and Tangent Ratio of a Right Triangle

Find and label the measure of each angle given two sides of the right triangle.

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