### Understanding Trigonometry in Right Triangles #### Problem Statement: Given a right triangle \( \triangle LJK \) with the right angle at \( K \), side \( LJ \) measuring \( 10 \) inches, side \( KJ \) measuring \( 15 \) inches, and the hypotenuse \( LK \) denoted as \( x \), determine the correct trigonometric equation for angle \( LJK \). #### Diagram: A right triangle \( \triangle LJK \) is depicted with a right angle at \( K \). The sides are labeled as follows: - \( LJ = 10 \) inches (adjacent to angle \( LJK \)) - \( KJ = 15 \) inches (opposite to angle \( LJK \)) - \( \overline{LK} = x \) (hypotenuse) #### Multiple Choice Question: Which equation can be used to find the measure of angle \( LJK \)? - \( \sin(x) = \frac{10}{15} \) - \( \sin(x) = \frac{15}{10} \) - \( \cos(x) = \frac{10}{15} \) - \( \cos(x) = \frac{15}{10} \) #### Analysis: To find the measure of angle \( LJK \), we should use either the sine or cosine definition. - **Sine of an angle** is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] - For angle \( LJK \), this would be \( \sin(LJK) = \frac{15}{x} \). - **Cosine of an angle** is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] - For angle \( LJK \), this would be \( \cos(LJK) = \frac{10}{x} \). However, in the options provided, we need to match the actual side lengths to their trigonometric counterparts without involving \( x \) explicitly. #### Explanation: - **Option 1
### Understanding Trigonometry in Right Triangles #### Problem Statement: Given a right triangle \( \triangle LJK \) with the right angle at \( K \), side \( LJ \) measuring \( 10 \) inches, side \( KJ \) measuring \( 15 \) inches, and the hypotenuse \( LK \) denoted as \( x \), determine the correct trigonometric equation for angle \( LJK \). #### Diagram: A right triangle \( \triangle LJK \) is depicted with a right angle at \( K \). The sides are labeled as follows: - \( LJ = 10 \) inches (adjacent to angle \( LJK \)) - \( KJ = 15 \) inches (opposite to angle \( LJK \)) - \( \overline{LK} = x \) (hypotenuse) #### Multiple Choice Question: Which equation can be used to find the measure of angle \( LJK \)? - \( \sin(x) = \frac{10}{15} \) - \( \sin(x) = \frac{15}{10} \) - \( \cos(x) = \frac{10}{15} \) - \( \cos(x) = \frac{15}{10} \) #### Analysis: To find the measure of angle \( LJK \), we should use either the sine or cosine definition. - **Sine of an angle** is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] - For angle \( LJK \), this would be \( \sin(LJK) = \frac{15}{x} \). - **Cosine of an angle** is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] - For angle \( LJK \), this would be \( \cos(LJK) = \frac{10}{x} \). However, in the options provided, we need to match the actual side lengths to their trigonometric counterparts without involving \( x \) explicitly. #### Explanation: - **Option 1
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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![### Understanding Trigonometry in Right Triangles
#### Problem Statement:
Given a right triangle \( \triangle LJK \) with the right angle at \( K \), side \( LJ \) measuring \( 10 \) inches, side \( KJ \) measuring \( 15 \) inches, and the hypotenuse \( LK \) denoted as \( x \), determine the correct trigonometric equation for angle \( LJK \).
#### Diagram:
A right triangle \( \triangle LJK \) is depicted with a right angle at \( K \). The sides are labeled as follows:
- \( LJ = 10 \) inches (adjacent to angle \( LJK \))
- \( KJ = 15 \) inches (opposite to angle \( LJK \))
- \( \overline{LK} = x \) (hypotenuse)
#### Multiple Choice Question:
Which equation can be used to find the measure of angle \( LJK \)?
- \( \sin(x) = \frac{10}{15} \)
- \( \sin(x) = \frac{15}{10} \)
- \( \cos(x) = \frac{10}{15} \)
- \( \cos(x) = \frac{15}{10} \)
#### Analysis:
To find the measure of angle \( LJK \), we should use either the sine or cosine definition.
- **Sine of an angle** is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
- For angle \( LJK \), this would be \( \sin(LJK) = \frac{15}{x} \).
- **Cosine of an angle** is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
- For angle \( LJK \), this would be \( \cos(LJK) = \frac{10}{x} \).
However, in the options provided, we need to match the actual side lengths to their trigonometric counterparts without involving \( x \) explicitly.
#### Explanation:
- **Option 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fefc4e4-0fc1-4d3a-92e0-1dd9cbc9acda%2Ff6c403cc-5545-4ee3-bda9-6d01c2270a85%2Fyrca8ec.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Trigonometry in Right Triangles
#### Problem Statement:
Given a right triangle \( \triangle LJK \) with the right angle at \( K \), side \( LJ \) measuring \( 10 \) inches, side \( KJ \) measuring \( 15 \) inches, and the hypotenuse \( LK \) denoted as \( x \), determine the correct trigonometric equation for angle \( LJK \).
#### Diagram:
A right triangle \( \triangle LJK \) is depicted with a right angle at \( K \). The sides are labeled as follows:
- \( LJ = 10 \) inches (adjacent to angle \( LJK \))
- \( KJ = 15 \) inches (opposite to angle \( LJK \))
- \( \overline{LK} = x \) (hypotenuse)
#### Multiple Choice Question:
Which equation can be used to find the measure of angle \( LJK \)?
- \( \sin(x) = \frac{10}{15} \)
- \( \sin(x) = \frac{15}{10} \)
- \( \cos(x) = \frac{10}{15} \)
- \( \cos(x) = \frac{15}{10} \)
#### Analysis:
To find the measure of angle \( LJK \), we should use either the sine or cosine definition.
- **Sine of an angle** is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
- For angle \( LJK \), this would be \( \sin(LJK) = \frac{15}{x} \).
- **Cosine of an angle** is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
- For angle \( LJK \), this would be \( \cos(LJK) = \frac{10}{x} \).
However, in the options provided, we need to match the actual side lengths to their trigonometric counterparts without involving \( x \) explicitly.
#### Explanation:
- **Option 1
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