Use an inverse trigonometric function to write 0 as a function of x. 0 = X 2
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°.
Related questions
Question
![### Using Inverse Trigonometric Functions to Express θ as a Function of x
**Problem Statement:**
Use an inverse trigonometric function to write \( \theta \) as a function of \( x \).
\[ \theta = \boxed{\text{}} \]
**Diagram Description:**
The diagram shows a right triangle with:
- The angle \( \theta \) located at the bottom left corner.
- The side opposite to \( \theta \) labeled as \( x \).
- The adjacent side to \( \theta \) labeled as \( 2 \).
- The right-angle marked in red.
### Solution
To solve for \( \theta \) using inverse trigonometric functions, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.
The formula is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is \( x \) and the adjacent side is \( 2 \). So,
\[ \tan(\theta) = \frac{x}{2} \]
To find \( \theta \), you take the inverse tangent (arctangent) of both sides:
\[ \theta = \tan^{-1}\left( \frac{x}{2} \right) \]
Therefore, the expression for \( \theta \) as a function of \( x \) is:
\[ \boxed{\tan^{-1}\left( \frac{x}{2} \right)} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F8cc09cd0-a3b0-4ddc-ad83-50b840db5064%2Fc374qxi_processed.png&w=3840&q=75)
Transcribed Image Text:### Using Inverse Trigonometric Functions to Express θ as a Function of x
**Problem Statement:**
Use an inverse trigonometric function to write \( \theta \) as a function of \( x \).
\[ \theta = \boxed{\text{}} \]
**Diagram Description:**
The diagram shows a right triangle with:
- The angle \( \theta \) located at the bottom left corner.
- The side opposite to \( \theta \) labeled as \( x \).
- The adjacent side to \( \theta \) labeled as \( 2 \).
- The right-angle marked in red.
### Solution
To solve for \( \theta \) using inverse trigonometric functions, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.
The formula is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is \( x \) and the adjacent side is \( 2 \). So,
\[ \tan(\theta) = \frac{x}{2} \]
To find \( \theta \), you take the inverse tangent (arctangent) of both sides:
\[ \theta = \tan^{-1}\left( \frac{x}{2} \right) \]
Therefore, the expression for \( \theta \) as a function of \( x \) is:
\[ \boxed{\tan^{-1}\left( \frac{x}{2} \right)} \]
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