 Find the exact value of each expression, if it is defined. (If an answer is undefined, enter UNDEFINED.)
(a)
sin−1(−1)
(b)
sin−1 
	2
2
(c)
sin−1(−5)


To solve these, let's recall that the function \( \sin^{-1}(x) \), or the inverse sine function (arcsin), is defined only for values \( x \) in the range \([-1, 1]\). The output (or range) of \( \sin^{-1}(x) \) is restricted to \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

### Solution

**(a) \( \sin^{-1}(-1) \)**

Since \( -1 \) is within the range \([-1, 1]\), we can evaluate it:
\[
\sin^{-1}(-1) = -\frac{\pi}{2}
\]
This is because \( \sin\left(-\frac{\pi}{2}\right) = -1 \).

**Answer:** \( -\frac{\pi}{2} \)

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**(b) \( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \)**

Since \( \frac{\sqrt{2}}{2} \) is within the range \([-1, 1]\), we can evaluate it:
\[
\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}
\]
This is because \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

**Answer:** \( \frac{\pi}{4} \)

---

**(c) \( \sin^{-1}(-5) \)**

Since \( -5 \) is outside the range \([-1, 1]\), \( \sin^{-1}(-5) \) is undefined.

**Answer:** UNDEFINED

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Would you like a more detailed explanation for any of these solutions?

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### Related Questions

1. What is the exact value of \( \sin^{-1}(1) \)?
2. How would you evaluate \( \cos^{-1}(-1) \)?
3. Why is the domain of \( \sin^{-1}(x) \) limited to \([-1, 1]\)?
4. How do you evaluate \( \sin\left(\sin^{-1}(x)\right) \) for \( x \) within \([-1, 1]\)?
5. What is the value of \( \tan^{-1}(1) \) in terms of \( \pi \)?

### Tip

When working with inverse trigonometric functions, always check if the input lies within the domain, as values outside of the defined range will be **undefined**.

Find the exact value of each expression, if it is defined. (If an answer is undefined, enter UNDEFINED.)
(a) sin−1(−1)
(b) sin−1(√2/2)
(c) sin−1(−5)

Solve for the exact values of inverse sine expressions, including sin−1(−1), sin−1(√2/2), and sin−1(−5). This problem covers inverse trigonometric functions, domain restrictions, and evaluations. Ideal for high school trigonometry students in Grades 10-12.

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