For the right triangle below, find the length of x. <-- 36 ° 5
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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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.
Question
![### Finding the Length of a Side in a Right Triangle
In the given right triangle, we are tasked with finding the length of side \( x \).
#### Triangle Information:
- One angle of the triangle is \( 36^\circ \).
- The side opposite this angle (\( x \)) is marked in the diagram.
- The length of the side adjacent to the angle, which is also the base of the right triangle, is labeled as 5 units.
To find the length of side \( x \), we can use the trigonometric function tangent, which is defined for an angle \(\theta\) in a right triangle as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this problem:
- \(\theta = 36^\circ\)
- The opposite side is \( x \)
- The adjacent side is \( 5 \)
Using the tangent function, we can set up the equation:
\[ \tan(36^\circ) = \frac{x}{5} \]
To solve for \( x \):
\[ x = 5 \times \tan(36^\circ) \]
### Calculation:
\[ \tan(36^\circ) \approx 0.7265 \]
Therefore:
\[ x = 5 \times 0.7265 \approx 3.6325 \]
So, the length of side \( x \) is approximately 3.6325 units.
---
### Diagram Explanation:
- The triangle is a right triangle (one angle is \( 90^\circ \)).
- One non-right angle is \( 36^\circ \).
- The side opposite the \( 36^\circ \) angle is labeled \( x \).
- The side adjacent to the \( 36^\circ \) angle is labeled 5 units.
- There is a small box at the corner of the triangle indicating the right angle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a49397d-2847-420e-bdfe-93d5dc178c48%2Fd4288c89-1be0-434d-921f-a5bf961a90a9%2Fja56j0r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Length of a Side in a Right Triangle
In the given right triangle, we are tasked with finding the length of side \( x \).
#### Triangle Information:
- One angle of the triangle is \( 36^\circ \).
- The side opposite this angle (\( x \)) is marked in the diagram.
- The length of the side adjacent to the angle, which is also the base of the right triangle, is labeled as 5 units.
To find the length of side \( x \), we can use the trigonometric function tangent, which is defined for an angle \(\theta\) in a right triangle as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this problem:
- \(\theta = 36^\circ\)
- The opposite side is \( x \)
- The adjacent side is \( 5 \)
Using the tangent function, we can set up the equation:
\[ \tan(36^\circ) = \frac{x}{5} \]
To solve for \( x \):
\[ x = 5 \times \tan(36^\circ) \]
### Calculation:
\[ \tan(36^\circ) \approx 0.7265 \]
Therefore:
\[ x = 5 \times 0.7265 \approx 3.6325 \]
So, the length of side \( x \) is approximately 3.6325 units.
---
### Diagram Explanation:
- The triangle is a right triangle (one angle is \( 90^\circ \)).
- One non-right angle is \( 36^\circ \).
- The side opposite the \( 36^\circ \) angle is labeled \( x \).
- The side adjacent to the \( 36^\circ \) angle is labeled 5 units.
- There is a small box at the corner of the triangle indicating the right angle.
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