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
![### Using Trigonometry to Find Angles
#### Problem:
Solve for \( x \). Round to the nearest tenth of a degree, if necessary.
#### Diagram:
The diagram depicts a triangle \( \Delta TUS \):
- **Vertices**: \( T \), \( U \), and \( S \)
- **Sides**:
- \( TU = 6.5 \) units (opposite side relative to \( \angle S \))
- \( US = 8.9 \) units (adjacent side to \( \angle T \))
- \( TS \) is the hypotenuse, unknown.
The angle at vertex \( S \) is labeled \( x \).
#### Instructions:
1. **Identify the appropriate trigonometric function**: Since the opposite (6.5) and adjacent sides (8.9) are known, use the tangent function which relates the angle to the ratio of the opposite side to the adjacent side:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6.5}{8.9}
\]
2. **Calculate the tangent ratio**:
\[
\tan(x) = \frac{6.5}{8.9} \approx 0.7303
\]
3. **Use the inverse tangent function to find the angle**:
\[
x = \tan^{-1}(0.7303)
\]
4. **Round the result to the nearest tenth of a degree, if necessary**.
By following these steps, you can determine the value of \( x \) using trigonometric principles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61c86ddb-00aa-4043-ac36-ebc52fad487f%2Fbedd23c9-3c1c-4ef1-88ca-f723a6cedb6b%2F2495rz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Trigonometry to Find Angles
#### Problem:
Solve for \( x \). Round to the nearest tenth of a degree, if necessary.
#### Diagram:
The diagram depicts a triangle \( \Delta TUS \):
- **Vertices**: \( T \), \( U \), and \( S \)
- **Sides**:
- \( TU = 6.5 \) units (opposite side relative to \( \angle S \))
- \( US = 8.9 \) units (adjacent side to \( \angle T \))
- \( TS \) is the hypotenuse, unknown.
The angle at vertex \( S \) is labeled \( x \).
#### Instructions:
1. **Identify the appropriate trigonometric function**: Since the opposite (6.5) and adjacent sides (8.9) are known, use the tangent function which relates the angle to the ratio of the opposite side to the adjacent side:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6.5}{8.9}
\]
2. **Calculate the tangent ratio**:
\[
\tan(x) = \frac{6.5}{8.9} \approx 0.7303
\]
3. **Use the inverse tangent function to find the angle**:
\[
x = \tan^{-1}(0.7303)
\]
4. **Round the result to the nearest tenth of a degree, if necessary**.
By following these steps, you can determine the value of \( x \) using trigonometric principles.
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