A 13 5 B 12

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
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### Finding the Measure of Angle BAC

In this exercise, we are given a triangle and asked to determine which equation can be used to find the measure of angle BAC.

#### Diagram Description
The triangle is a right triangle with the following dimensions:
- The side AC, which is adjacent to the angle BAC, is 12 units long.
- The side AB, which is the hypotenuse of the triangle, is 13 units long.
- The side BC, which is opposite to the angle BAC, is 5 units long.

#### Problem Statement
**Question:** Which equation can be used to find the measure of angle BAC?

Below are the options provided to solve for angle BAC:

1. \(\tan^{-1}\left(\frac{5}{12}\right) = x\)
2. \(\tan^{-1}\left(\frac{12}{5}\right) = x\)
3. \(\cos^{-1}\left(\frac{12}{13}\right) = x\)
4. \(\cos^{-1}\left(\frac{13}{12}\right) = x\)

#### Analysis
To determine the correct equation for finding the measure of angle BAC, we analyze the trigonometric functions involved. For a right triangle:

- **Tangent Function (tanθ):** The ratio of the opposite side to the adjacent side.
  \[
  \tan(BAC) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AC} = \frac{5}{12}
  \]

- **Cosine Function (cosθ):** The ratio of the adjacent side to the hypotenuse.
  \[
  \cos(BAC) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{12}{13}
  \]

By matching these ratios to the provided options, we can identify the correct equation used to find angle BAC. 

**Correct Answer:** 
The equations:
1. \(\tan^{-1}\left(\frac{5}{12}\right) = x\)
3. \(\cos^{-1}\left(\frac{12}{13}\right) = x\)

Both options can be used to find the measure of angle BAC.

It is essential for students to understand how to apply trigonometric ratios and inverse trigonometric functions to solve for angles in right triangles.
Transcribed Image Text:### Finding the Measure of Angle BAC In this exercise, we are given a triangle and asked to determine which equation can be used to find the measure of angle BAC. #### Diagram Description The triangle is a right triangle with the following dimensions: - The side AC, which is adjacent to the angle BAC, is 12 units long. - The side AB, which is the hypotenuse of the triangle, is 13 units long. - The side BC, which is opposite to the angle BAC, is 5 units long. #### Problem Statement **Question:** Which equation can be used to find the measure of angle BAC? Below are the options provided to solve for angle BAC: 1. \(\tan^{-1}\left(\frac{5}{12}\right) = x\) 2. \(\tan^{-1}\left(\frac{12}{5}\right) = x\) 3. \(\cos^{-1}\left(\frac{12}{13}\right) = x\) 4. \(\cos^{-1}\left(\frac{13}{12}\right) = x\) #### Analysis To determine the correct equation for finding the measure of angle BAC, we analyze the trigonometric functions involved. For a right triangle: - **Tangent Function (tanθ):** The ratio of the opposite side to the adjacent side. \[ \tan(BAC) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AC} = \frac{5}{12} \] - **Cosine Function (cosθ):** The ratio of the adjacent side to the hypotenuse. \[ \cos(BAC) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{12}{13} \] By matching these ratios to the provided options, we can identify the correct equation used to find angle BAC. **Correct Answer:** The equations: 1. \(\tan^{-1}\left(\frac{5}{12}\right) = x\) 3. \(\cos^{-1}\left(\frac{12}{13}\right) = x\) Both options can be used to find the measure of angle BAC. It is essential for students to understand how to apply trigonometric ratios and inverse trigonometric functions to solve for angles in right triangles.
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