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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Question
![### Triangle Trigonometry Problem
**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram:**
The provided diagram is a right triangle \( KJI \). Here are the key details:
- Angle \( \angle KJI \) is a right angle (90°).
- Angle \( \angle KIJ \) is given as 63°.
- The length of the side opposite the 63° angle, \( IJ \), is 1 unit.
- The length \( x \) is the side adjacent to the 63° angle, between points \( K \) and \( J \).
**Solution:**
To solve for \( x \), we can use the tangent function, which relates the angle of a right triangle to the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For this problem, the angle \(\theta\) is 63°, the opposite side is 1 unit, and the adjacent side is \( x \).
\[ \tan(63°) = \frac{1}{x} \]
Solve for \( x \):
\[ x = \frac{1}{\tan(63°)} \]
Using a calculator to find \(\tan(63°)\):
\[ \tan(63°) \approx 1.9626 \]
\[ x = \frac{1}{1.9626} \approx 0.5096 \]
Rounding to the nearest tenth:
\[ x \approx 0.5 \]
Hence, the length \( x \) is approximately 0.5 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08807450-21ed-4e55-84b1-e841f0af059c%2F600bbb36-5613-4ccd-aad0-1b001903dc47%2Fnrvlhz9_processed.png&w=3840&q=75)
Transcribed Image Text:### Triangle Trigonometry Problem
**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram:**
The provided diagram is a right triangle \( KJI \). Here are the key details:
- Angle \( \angle KJI \) is a right angle (90°).
- Angle \( \angle KIJ \) is given as 63°.
- The length of the side opposite the 63° angle, \( IJ \), is 1 unit.
- The length \( x \) is the side adjacent to the 63° angle, between points \( K \) and \( J \).
**Solution:**
To solve for \( x \), we can use the tangent function, which relates the angle of a right triangle to the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For this problem, the angle \(\theta\) is 63°, the opposite side is 1 unit, and the adjacent side is \( x \).
\[ \tan(63°) = \frac{1}{x} \]
Solve for \( x \):
\[ x = \frac{1}{\tan(63°)} \]
Using a calculator to find \(\tan(63°)\):
\[ \tan(63°) \approx 1.9626 \]
\[ x = \frac{1}{1.9626} \approx 0.5096 \]
Rounding to the nearest tenth:
\[ x \approx 0.5 \]
Hence, the length \( x \) is approximately 0.5 units.
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