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
Problem 1CT
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Topic Video
Question
![The image features a trigonometry problem involving a right triangle and multiple-choice answers to identify which trigonometric equation can be used to find the measure of angle BAC.
### Description of the Right Triangle Diagram:
- **Vertices**: The triangle has vertices labeled A, B, and C.
- **Sides**:
- The side opposite to angle BAC is labeled 5 units.
- The side adjacent to angle BAC is labeled 12 units.
- The hypotenuse (the side opposite the right angle at vertex C) is labeled 13 units.
### Diagram Explanation:
- **Point A** is at the top vertex of the triangle.
- **Point B** is at the bottom-right vertex.
- **Point C** is at the bottom-left vertex and indicates the right angle with a small red square.
- **Line Segment AC** is the side opposite angle BAC and is 5 units long.
- **Line Segment CB** is the side adjacent to angle BAC and is 12 units long.
- **Line Segment AB** is the hypotenuse of the right triangle and is 13 units long.
### Question:
"Which equation can be used to find the measure of angle BAC?"
### Answer Choices:
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. \(\cos^{-1} \left(\frac{13}{12}\right) = x\)
### Detailed Explanation:
To determine the correct equation for finding the measure of angle BAC, we need to consider the trigonometric ratios associated with a right triangle:
- **Tangent (tan)**:
\[
\tan(\text{{angle}}) = \frac{\text{{opposite}}}{\text{{adjacent}}}
\]
For angle BAC, the opposite side is 5 units and the adjacent side is 12 units. Thus, the correct tangent-based equation is:
\[
\tan^{-1} \left(\frac{5}{12}\right) = x
\]
- **Cosine (cos)**:
\[
\cos(\text{{angle}}) = \frac{\text{{adj](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fefc4e4-0fc1-4d3a-92e0-1dd9cbc9acda%2F342d1d21-920d-417c-83b4-32d4cbff1efc%2F7b7fyh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image features a trigonometry problem involving a right triangle and multiple-choice answers to identify which trigonometric equation can be used to find the measure of angle BAC.
### Description of the Right Triangle Diagram:
- **Vertices**: The triangle has vertices labeled A, B, and C.
- **Sides**:
- The side opposite to angle BAC is labeled 5 units.
- The side adjacent to angle BAC is labeled 12 units.
- The hypotenuse (the side opposite the right angle at vertex C) is labeled 13 units.
### Diagram Explanation:
- **Point A** is at the top vertex of the triangle.
- **Point B** is at the bottom-right vertex.
- **Point C** is at the bottom-left vertex and indicates the right angle with a small red square.
- **Line Segment AC** is the side opposite angle BAC and is 5 units long.
- **Line Segment CB** is the side adjacent to angle BAC and is 12 units long.
- **Line Segment AB** is the hypotenuse of the right triangle and is 13 units long.
### Question:
"Which equation can be used to find the measure of angle BAC?"
### Answer Choices:
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. \(\cos^{-1} \left(\frac{13}{12}\right) = x\)
### Detailed Explanation:
To determine the correct equation for finding the measure of angle BAC, we need to consider the trigonometric ratios associated with a right triangle:
- **Tangent (tan)**:
\[
\tan(\text{{angle}}) = \frac{\text{{opposite}}}{\text{{adjacent}}}
\]
For angle BAC, the opposite side is 5 units and the adjacent side is 12 units. Thus, the correct tangent-based equation is:
\[
\tan^{-1} \left(\frac{5}{12}\right) = x
\]
- **Cosine (cos)**:
\[
\cos(\text{{angle}}) = \frac{\text{{adj
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