Find the exact value of the real number y in radians. y = tan-1 (-1) %3D
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 the real number \( y \) in radians.**
\[ y = \tan^{-1}(-1) \]
### Explanation
This problem requires finding the inverse tangent, or arctangent, of \(-1\). The arctangent function, \(\tan^{-1}(x)\), returns the angle whose tangent is \(x\). The result is typically given in radians within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
For \(-1\), we seek an angle \(\theta\) such that:
\[ \tan(\theta) = -1 \]
In the unit circle, this corresponds to an angle of \(-\frac{\pi}{4}\) radians, as \(\tan(-\frac{\pi}{4}) = -1\).
Therefore, the exact value of \( y \) is:
\[ y = -\frac{\pi}{4} \]
### Additional Notes
- No graphs or additional diagrams are included in the problem. Understanding this requires familiarity with trigonometric functions and the unit circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb592224-c5dc-4690-b44a-e8ba296a38eb%2F0cc2ab16-6f6c-4ce1-bc18-9c6e316c6442%2Fgonwpro_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Find the exact value of the real number \( y \) in radians.**
\[ y = \tan^{-1}(-1) \]
### Explanation
This problem requires finding the inverse tangent, or arctangent, of \(-1\). The arctangent function, \(\tan^{-1}(x)\), returns the angle whose tangent is \(x\). The result is typically given in radians within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
For \(-1\), we seek an angle \(\theta\) such that:
\[ \tan(\theta) = -1 \]
In the unit circle, this corresponds to an angle of \(-\frac{\pi}{4}\) radians, as \(\tan(-\frac{\pi}{4}) = -1\).
Therefore, the exact value of \( y \) is:
\[ y = -\frac{\pi}{4} \]
### Additional Notes
- No graphs or additional diagrams are included in the problem. Understanding this requires familiarity with trigonometric functions and the unit circle.
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