Find the exact value of y or state that y is undefined. y=sin [tan ¹(-8)]
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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![**Problem Statement:**
Find the exact value of \( y \) or state that \( y \) is undefined.
\[ y = \sin \left[ \tan^{-1}(-8) \right] \]
**Explanation:**
This problem requires finding the sine of the angle whose tangent is \(-8\). To solve this, consider the following:
1. **Inverse Tangent Function (\(\tan^{-1}\)):**
- The angle \( \theta = \tan^{-1}(-8) \) implies that the tangent of \( \theta \) is \(-8\).
- Imagine a right triangle where the opposite side is \(-8\) and the adjacent side is \(1\) (or vice versa for angles in different quadrants).
2. **Pythagorean Theorem:**
- To find the hypotenuse, use the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{(-8)^2 + 1^2} = \sqrt{64 + 1} = \sqrt{65}
\]
3. **Calculating the Sine:**
- Sine is the ratio of the opposite side to the hypotenuse. Therefore, \(\sin(\theta) = \frac{-8}{\sqrt{65}}\).
4. **Final Expression:**
- \( y = \sin \left[ \tan^{-1}(-8) \right] = \frac{-8}{\sqrt{65}} \).
This computation gives the exact value of \( y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcea6d79-c0c2-49cd-aa31-c3eca199518f%2F7e7fd1a2-6e1d-4c0a-9dc8-c07889421396%2Fwvwkwxr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the exact value of \( y \) or state that \( y \) is undefined.
\[ y = \sin \left[ \tan^{-1}(-8) \right] \]
**Explanation:**
This problem requires finding the sine of the angle whose tangent is \(-8\). To solve this, consider the following:
1. **Inverse Tangent Function (\(\tan^{-1}\)):**
- The angle \( \theta = \tan^{-1}(-8) \) implies that the tangent of \( \theta \) is \(-8\).
- Imagine a right triangle where the opposite side is \(-8\) and the adjacent side is \(1\) (or vice versa for angles in different quadrants).
2. **Pythagorean Theorem:**
- To find the hypotenuse, use the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{(-8)^2 + 1^2} = \sqrt{64 + 1} = \sqrt{65}
\]
3. **Calculating the Sine:**
- Sine is the ratio of the opposite side to the hypotenuse. Therefore, \(\sin(\theta) = \frac{-8}{\sqrt{65}}\).
4. **Final Expression:**
- \( y = \sin \left[ \tan^{-1}(-8) \right] = \frac{-8}{\sqrt{65}} \).
This computation gives the exact value of \( y \).
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