Evaluate the expression.
$\(\sin\)^{-1}\(\left\)(\(\frac{\sqrt2}{2}\]\right\))$

• A. $\frac{\pi }{4}$
• B. $-\frac{\pi }{4}$
• C. $\frac{3\pi }{4}$
• D. $\frac{7\pi }{4}$

Answer: (A)

Verified step by step guidance

1. Understand that sin⁡−1(x) or arcsin(x) is the inverse function of sin(x), which means it gives the angle whose sine is x.
2. Recognize that the expression sin⁡−1(√2/2) asks for the angle θ such that sin(θ) = √2/2.
3. Recall that the sine of π/4 (or 45 degrees) is √2/2. Therefore, one possible angle is θ = π/4.
4. Consider the range of the arcsin function, which is [-π/2, π/2]. This means the principal value of arcsin(√2/2) is within this range.
5. Conclude that the angle θ = π/4 is the correct answer since it lies within the range of the arcsin function and satisfies sin(θ) = √2/2.