Find the angle 0, in radians, in the given right triangle. The length of the side adjacent to 0 is 18 and the length of the side opposite 0 is 15. 15 18 Round your answer to the nearest hundredth.
Find the angle 0, in radians, in the given right triangle. The length of the side adjacent to 0 is 18 and the length of the side opposite 0 is 15. 15 18 Round your answer to the nearest hundredth.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find the angle \( \theta \) in radians, in the given right triangle. The length of the side adjacent to \( \theta \) is 18 and the length of the side opposite \( \theta \) is 15.
*Round your answer to the nearest hundredth.*
**Diagram Explanation:**
The provided diagram is a right triangle, and it is marked as follows:
- \( \theta \) is one of the angles in the triangle.
- The side adjacent to \( \theta \) is labeled with a length of 18.
- The side opposite \( \theta \) is labeled with a length of 15.
- There is a right angle indicated by the square in the corner opposite the hypotenuse.
**Solution Approach:**
To find the angle \( \theta \) in radians, you can use the tangent function, which relates the opposite side and the adjacent side in a right triangle. The tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side.
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- Opposite side length = 15
- Adjacent side length = 18
Using this ratio,
\[ \tan(\theta) = \frac{15}{18} = \frac{5}{6} \]
To find \( \theta \), take the arctangent (inverse tangent) of \( \frac{5}{6} \):
\[ \theta = \arctan \left(\frac{5}{6}\right) \]
Calculating this in radians:
\[ \theta \approx 0.6947 \]
So, the angle \( \theta \) is approximately 0.69 radians when rounded to the nearest hundredth.
**Conclusion:**
By using trigonometric functions and the properties of a right triangle, we determined that the angle \( \theta \) in the given right triangle is approximately 0.69 radians.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F704f324a-66c2-47cf-962c-2947999b0de6%2F42c2148b-b840-4fc0-9ac2-1ced46fcab21%2Fa1be1f9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the angle \( \theta \) in radians, in the given right triangle. The length of the side adjacent to \( \theta \) is 18 and the length of the side opposite \( \theta \) is 15.
*Round your answer to the nearest hundredth.*
**Diagram Explanation:**
The provided diagram is a right triangle, and it is marked as follows:
- \( \theta \) is one of the angles in the triangle.
- The side adjacent to \( \theta \) is labeled with a length of 18.
- The side opposite \( \theta \) is labeled with a length of 15.
- There is a right angle indicated by the square in the corner opposite the hypotenuse.
**Solution Approach:**
To find the angle \( \theta \) in radians, you can use the tangent function, which relates the opposite side and the adjacent side in a right triangle. The tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side.
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- Opposite side length = 15
- Adjacent side length = 18
Using this ratio,
\[ \tan(\theta) = \frac{15}{18} = \frac{5}{6} \]
To find \( \theta \), take the arctangent (inverse tangent) of \( \frac{5}{6} \):
\[ \theta = \arctan \left(\frac{5}{6}\right) \]
Calculating this in radians:
\[ \theta \approx 0.6947 \]
So, the angle \( \theta \) is approximately 0.69 radians when rounded to the nearest hundredth.
**Conclusion:**
By using trigonometric functions and the properties of a right triangle, we determined that the angle \( \theta \) in the given right triangle is approximately 0.69 radians.
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