sin sin 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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Sin^-1(sin 9pi/8)=
![The image contains a mathematical expression involving trigonometric functions and their inverses. The expression is:
\[ \sin^{-1} \left( \sin \frac{9\pi}{8} \right) \]
This can be interpreted as finding the inverse sine (also known as arc sine, denoted as \(\sin^{-1}\)) of the sine of \( \frac{9\pi}{8} \).
### Steps to Simplify:
1. **Sine Function**:
- Evaluate \( \sin \left( \frac{9\pi}{8} \right) \):
- Since \( \frac{9\pi}{8} \) is more than \( \pi \) (which is approximately 3.14), it is in the third quadrant of the unit circle where sine values are negative.
- As \( \frac{9\pi}{8} \) exactly equals \( \pi + \frac{\pi}{8} \), we can rewrite sine as: \( \sin \left( \pi + \frac{\pi}{8} \right) \).
- Using the property of sine \(\sin(\pi + x) = -\sin(x)\), it follows:
\[ \sin \left( \frac{9\pi}{8} \right) = -\sin \left( \frac{\pi}{8} \right) \]
2. **Inverse Sine Function**:
- Now, find \( \sin^{-1} \left( -\sin \left( \frac{\pi}{8} \right) \right) \):
- Inverse sine function will give us an angle whose sine value is \( -\sin \left( \frac{\pi}{8} \right) \). Since the inverse sine function will give an angle in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), in this case the value will be:
\[ \sin^{-1} \left( -\sin \left( \frac{\pi}{8} \right) \right) = - \frac{\pi}{8} \]
### Conclusion:
So, the simplified form of the expression \( \sin^{-1} \left( \sin \frac{9\pi}{8} \right) \) is \( - \frac{\pi}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b89a6c-6ca3-4f25-9a44-1409c69966fc%2Fa69f4a67-b192-4063-ace9-8fb3f1f4de02%2Ffom3a6w_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression involving trigonometric functions and their inverses. The expression is:
\[ \sin^{-1} \left( \sin \frac{9\pi}{8} \right) \]
This can be interpreted as finding the inverse sine (also known as arc sine, denoted as \(\sin^{-1}\)) of the sine of \( \frac{9\pi}{8} \).
### Steps to Simplify:
1. **Sine Function**:
- Evaluate \( \sin \left( \frac{9\pi}{8} \right) \):
- Since \( \frac{9\pi}{8} \) is more than \( \pi \) (which is approximately 3.14), it is in the third quadrant of the unit circle where sine values are negative.
- As \( \frac{9\pi}{8} \) exactly equals \( \pi + \frac{\pi}{8} \), we can rewrite sine as: \( \sin \left( \pi + \frac{\pi}{8} \right) \).
- Using the property of sine \(\sin(\pi + x) = -\sin(x)\), it follows:
\[ \sin \left( \frac{9\pi}{8} \right) = -\sin \left( \frac{\pi}{8} \right) \]
2. **Inverse Sine Function**:
- Now, find \( \sin^{-1} \left( -\sin \left( \frac{\pi}{8} \right) \right) \):
- Inverse sine function will give us an angle whose sine value is \( -\sin \left( \frac{\pi}{8} \right) \). Since the inverse sine function will give an angle in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), in this case the value will be:
\[ \sin^{-1} \left( -\sin \left( \frac{\pi}{8} \right) \right) = - \frac{\pi}{8} \]
### Conclusion:
So, the simplified form of the expression \( \sin^{-1} \left( \sin \frac{9\pi}{8} \right) \) is \( - \frac{\pi}{
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