Find the exact value, if any, of the following composite function. 元 sin sin
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 5GP
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![**Topic: Evaluating Composite Trigonometric Functions**
**Question:**
Find the exact value, if any, of the following composite function. Do not use a calculator.
\[ \sin^{-1} \left( \sin \left( \frac{7\pi}{5} \right) \right) \]
**Instructions:**
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. \[ \sin^{-1} \left( \sin \left( \frac{7\pi}{5} \right) \right) = \]
(Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.)
B. It is not defined.
**Explanation:**
To solve this, you need to consider the properties and ranges of the sine and inverse sine functions. Remember:
- The sine function, \(\sin(x)\), is periodic with a period of \(2\pi\).
- The inverse sine function, \(\sin^{-1}(x)\), also called arcsine, has a range of \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This means that the output of the inverse sine function must lie within this interval.
1. Determine if \( \frac{7\pi}{5} \) fits within the range of \( \sin^{-1}(x) \):
- Since \( \frac{7\pi}{5} \) is outside the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we must find a coterminal angle that lies within this interval.
- Notice that \( \frac{7\pi}{5} \) is greater than \( \pi \). Thus, consider the angle within the range of the sine inverse function by finding the equivalent angle.
2. Simplify \( \sin(\frac{7\pi}{5}) \):
- Observing periodicity, since sine function has a period of \(2\pi\):
\[ \frac{7\pi}{5} - 2\pi = \frac{7\pi}{5} - \frac{10\pi}{5} = -\frac{3\pi}{5} \]
- Now, \(-\frac{3\pi}{5}\) lies within the interval \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd018cc6-2cd8-4cb3-91a6-c6c76d6754a6%2F63f8e617-78e0-4b1d-ad17-c949b7c90035%2F8rzxwts.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Evaluating Composite Trigonometric Functions**
**Question:**
Find the exact value, if any, of the following composite function. Do not use a calculator.
\[ \sin^{-1} \left( \sin \left( \frac{7\pi}{5} \right) \right) \]
**Instructions:**
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. \[ \sin^{-1} \left( \sin \left( \frac{7\pi}{5} \right) \right) = \]
(Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.)
B. It is not defined.
**Explanation:**
To solve this, you need to consider the properties and ranges of the sine and inverse sine functions. Remember:
- The sine function, \(\sin(x)\), is periodic with a period of \(2\pi\).
- The inverse sine function, \(\sin^{-1}(x)\), also called arcsine, has a range of \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This means that the output of the inverse sine function must lie within this interval.
1. Determine if \( \frac{7\pi}{5} \) fits within the range of \( \sin^{-1}(x) \):
- Since \( \frac{7\pi}{5} \) is outside the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we must find a coterminal angle that lies within this interval.
- Notice that \( \frac{7\pi}{5} \) is greater than \( \pi \). Thus, consider the angle within the range of the sine inverse function by finding the equivalent angle.
2. Simplify \( \sin(\frac{7\pi}{5}) \):
- Observing periodicity, since sine function has a period of \(2\pi\):
\[ \frac{7\pi}{5} - 2\pi = \frac{7\pi}{5} - \frac{10\pi}{5} = -\frac{3\pi}{5} \]
- Now, \(-\frac{3\pi}{5}\) lies within the interval \
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