Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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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}{
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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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