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
![**Educational Task: Solving for an Angle in a Right Triangle**
**Objective:**
Solve for the angle \( x \) in the given right triangle. Round your answer to the nearest tenth of a degree, if necessary.
**Diagram Explanation:**
There is a right triangle labeled \( \triangle FGH \) with the right angle at \( G \). The sides of the triangle are labeled as follows:
- The side opposite the right angle (hypotenuse \( FH \)) is \( 1.6 \) units.
- The side opposite angle \( x \) (side \( GH \)) is \( 1.2 \) units.
**Steps to Solve:**
1. **Identify Known Values:**
- Hypotenuse (\( FH \)) = 1.6 units
- Opposite side (\( GH \)) = 1.2 units
2. **Trigonometric Ratio:**
Use the sine function, which relates the opposite side to the hypotenuse in a right triangle.
\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
3. **Substitute Known Values:**
\[
\sin(x) = \frac{1.2}{1.6}
\]
4. **Calculate the Ratio:**
\[
\sin(x) = 0.75
\]
5. **Solve for \( x \):**
Take the inverse sine (arcsine) of 0.75 to find the angle \( x \).
\[
x = \sin^{-1}(0.75)
\]
6. **Use a Calculator:**
Using a scientific calculator, you find that
\[
x \approx 48.6^\circ
\]
**Answer:**
The angle \( x \) is approximately \( 48.6^\circ \) when rounded to the nearest tenth of a degree.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5406ca01-59af-4525-aa74-c4d606a1cb55%2Fa792d6a9-b151-4ac8-8390-b0dc0513f225%2Fgg9an1rw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Task: Solving for an Angle in a Right Triangle**
**Objective:**
Solve for the angle \( x \) in the given right triangle. Round your answer to the nearest tenth of a degree, if necessary.
**Diagram Explanation:**
There is a right triangle labeled \( \triangle FGH \) with the right angle at \( G \). The sides of the triangle are labeled as follows:
- The side opposite the right angle (hypotenuse \( FH \)) is \( 1.6 \) units.
- The side opposite angle \( x \) (side \( GH \)) is \( 1.2 \) units.
**Steps to Solve:**
1. **Identify Known Values:**
- Hypotenuse (\( FH \)) = 1.6 units
- Opposite side (\( GH \)) = 1.2 units
2. **Trigonometric Ratio:**
Use the sine function, which relates the opposite side to the hypotenuse in a right triangle.
\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
3. **Substitute Known Values:**
\[
\sin(x) = \frac{1.2}{1.6}
\]
4. **Calculate the Ratio:**
\[
\sin(x) = 0.75
\]
5. **Solve for \( x \):**
Take the inverse sine (arcsine) of 0.75 to find the angle \( x \).
\[
x = \sin^{-1}(0.75)
\]
6. **Use a Calculator:**
Using a scientific calculator, you find that
\[
x \approx 48.6^\circ
\]
**Answer:**
The angle \( x \) is approximately \( 48.6^\circ \) when rounded to the nearest tenth of a degree.
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