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
Related questions
Question
![### Solving for \( x \) in a Right Triangle
#### Problem Statement:
Solve for \( x \) in the triangle. Round your answer to the nearest tenth.
#### Diagram Description:
A right triangle is given with the following characteristics:
- One angle is \( 27^\circ \).
- The length of the side adjacent to the \( 27^\circ \) angle is \( 5 \) units.
- The hypotenuse is labeled \( x \).
- The right angle is opposite the hypotenuse.
#### Solution Steps:
1. **Identify the Trigonometric Function**:
For a right triangle, to find the hypotenuse \( x \) when you have the adjacent side and an angle, you use the cosine ratio:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Here, \(\theta = 27^\circ\), the adjacent side is \( 5 \), and the hypotenuse is \( x \).
2. **Set Up the Equation**:
\[
\cos(27^\circ) = \frac{5}{x}
\]
3. **Solve for \( x \)**:
\[
x = \frac{5}{\cos(27^\circ)}
\]
Use a calculator to find \(\cos(27^\circ)\):
4. **Calculate**:
\[
\cos(27^\circ) \approx 0.8910
\]
Plugging this value back into the equation gives:
\[
x \approx \frac{5}{0.8910} \approx 5.6
\]
5. **Round the Answer**:
The rounded value of \( x \) is \( 5.6 \).
#### Final Answer:
\[
x \approx 5.6
\]
For further practice, make sure to review trigonometric functions and their applications in solving right triangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd1ff857-92ff-4e51-81f6-c8581968700f%2F3f5265bc-b1d9-4b71-9ca5-c225ad545879%2Fdh6f6el_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for \( x \) in a Right Triangle
#### Problem Statement:
Solve for \( x \) in the triangle. Round your answer to the nearest tenth.
#### Diagram Description:
A right triangle is given with the following characteristics:
- One angle is \( 27^\circ \).
- The length of the side adjacent to the \( 27^\circ \) angle is \( 5 \) units.
- The hypotenuse is labeled \( x \).
- The right angle is opposite the hypotenuse.
#### Solution Steps:
1. **Identify the Trigonometric Function**:
For a right triangle, to find the hypotenuse \( x \) when you have the adjacent side and an angle, you use the cosine ratio:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Here, \(\theta = 27^\circ\), the adjacent side is \( 5 \), and the hypotenuse is \( x \).
2. **Set Up the Equation**:
\[
\cos(27^\circ) = \frac{5}{x}
\]
3. **Solve for \( x \)**:
\[
x = \frac{5}{\cos(27^\circ)}
\]
Use a calculator to find \(\cos(27^\circ)\):
4. **Calculate**:
\[
\cos(27^\circ) \approx 0.8910
\]
Plugging this value back into the equation gives:
\[
x \approx \frac{5}{0.8910} \approx 5.6
\]
5. **Round the Answer**:
The rounded value of \( x \) is \( 5.6 \).
#### Final Answer:
\[
x \approx 5.6
\]
For further practice, make sure to review trigonometric functions and their applications in solving right triangles.
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