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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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![### Solving for x in a Right Triangle
**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Description:**
A right triangle \( \Delta GHI \) is given with the following attributes:
- \( GH \) = 4.5 units
- \( \angle HGI \) = \( 90^\circ \)
- Side \( x \) is opposite the angle \( \angle GIH = 17^\circ \)
The lengths of the sides and angles are labeled on the triangle as follows:
- \( \angle IHG = 17^\circ \)
- \( GH = 4.5 \) (the side adjacent to the 17° angle)
- \( x \) is the hypotenuse
Here is how you can solve for \( x \):
**Steps to Solution:**
To solve for \( x \), we can use trigonometric ratios, particularly the cosine function which is defined as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
In this triangle:
- The angle \( \theta = 17^\circ \)
- The adjacent side \( GH = 4.5 \)
- The hypotenuse is \( x \)
Using the cosine function:
\[ \cos(17^\circ) = \frac{4.5}{x} \]
Rearranging to solve for \( x \):
\[ x = \frac{4.5}{\cos(17^\circ)} \]
**Calculation:**
1. Find the cosine of 17 degrees using a calculator.
\[ \cos(17^\circ) \approx 0.9563 \]
2. Plug this value back into the equation:
\[ x = \frac{4.5}{0.9563} \approx 4.7 \]
**Solution:**
\[ x \approx 4.7 \]
Thus, the length of the hypotenuse \( x \) is approximately 4.7 units when rounded to the nearest tenth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17ee36ea-a061-4aba-8bad-366095b0251b%2F69a3efa9-73d6-4875-a6e0-03dd51d99851%2Fowmhcjg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for x in a Right Triangle
**Problem Statement:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Description:**
A right triangle \( \Delta GHI \) is given with the following attributes:
- \( GH \) = 4.5 units
- \( \angle HGI \) = \( 90^\circ \)
- Side \( x \) is opposite the angle \( \angle GIH = 17^\circ \)
The lengths of the sides and angles are labeled on the triangle as follows:
- \( \angle IHG = 17^\circ \)
- \( GH = 4.5 \) (the side adjacent to the 17° angle)
- \( x \) is the hypotenuse
Here is how you can solve for \( x \):
**Steps to Solution:**
To solve for \( x \), we can use trigonometric ratios, particularly the cosine function which is defined as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
In this triangle:
- The angle \( \theta = 17^\circ \)
- The adjacent side \( GH = 4.5 \)
- The hypotenuse is \( x \)
Using the cosine function:
\[ \cos(17^\circ) = \frac{4.5}{x} \]
Rearranging to solve for \( x \):
\[ x = \frac{4.5}{\cos(17^\circ)} \]
**Calculation:**
1. Find the cosine of 17 degrees using a calculator.
\[ \cos(17^\circ) \approx 0.9563 \]
2. Plug this value back into the equation:
\[ x = \frac{4.5}{0.9563} \approx 4.7 \]
**Solution:**
\[ x \approx 4.7 \]
Thus, the length of the hypotenuse \( x \) is approximately 4.7 units when rounded to the nearest tenth.
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