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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![**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Analysis:**
The provided diagram is a right triangle \( \triangle SQR \) with the following elements:
- \(\angle S\) is labeled as \(54^\circ\).
- The side \( SQ \) (opposite \(\angle S\)) is labeled as \(2.3\) units.
- The side \( RQ \) (adjacent to \(\angle S\)) is labeled as \( x \).
- \(\angle R\) is the right angle in the triangle.
**Explanation:**
Given the right triangle and the known angle, we can use trigonometric functions to find the value of \( x \).
1. Identify the relevant trigonometric ratio. Since we know the length of the opposite side (\(SQ\)) and need to find the length of the adjacent side (\(RQ\)), we will use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
2. Plug in the values:
\[
\tan(54^\circ) = \frac{2.3}{x}
\]
3. Solve for \( x \):
\[
x = \frac{2.3}{\tan(54^\circ)}
\]
4. Use a calculator to find \(\tan(54^\circ)\). The approximate value is:
\[
\tan(54^\circ) \approx 1.3764
\]
5. Calculate \( x \):
\[
x = \frac{2.3}{1.3764} \approx 1.67
\]
Therefore, \( x \approx 1.7 \) when rounded to the nearest tenth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb223a19b-0fd0-443c-881a-6f762159fad5%2F11520dd9-619e-4aaf-a237-43607513d91b%2F0bd58jj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Analysis:**
The provided diagram is a right triangle \( \triangle SQR \) with the following elements:
- \(\angle S\) is labeled as \(54^\circ\).
- The side \( SQ \) (opposite \(\angle S\)) is labeled as \(2.3\) units.
- The side \( RQ \) (adjacent to \(\angle S\)) is labeled as \( x \).
- \(\angle R\) is the right angle in the triangle.
**Explanation:**
Given the right triangle and the known angle, we can use trigonometric functions to find the value of \( x \).
1. Identify the relevant trigonometric ratio. Since we know the length of the opposite side (\(SQ\)) and need to find the length of the adjacent side (\(RQ\)), we will use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
2. Plug in the values:
\[
\tan(54^\circ) = \frac{2.3}{x}
\]
3. Solve for \( x \):
\[
x = \frac{2.3}{\tan(54^\circ)}
\]
4. Use a calculator to find \(\tan(54^\circ)\). The approximate value is:
\[
\tan(54^\circ) \approx 1.3764
\]
5. Calculate \( x \):
\[
x = \frac{2.3}{1.3764} \approx 1.67
\]
Therefore, \( x \approx 1.7 \) when rounded to the nearest tenth.
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