Problem 17 - Precalculus: A Concise Course

Question

Evaluate the expression without using a calculator. $$ \arcsin \frac{\sqrt{2}}{2} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{\pi}{4} \) or \( 45^\circ \).

Step 1: Identify Corresponding Angle

Remember that for arcsine, we're looking for the angle whose sine is \( \frac{\sqrt{2}}{2} \). From the knowledge of standard angles and unit circle, we know that the value of sine is \( \frac{\sqrt{2}}{2} \) when the angle is \( \frac{\pi}{4} \) or \( 45^\circ \)

Step 2: Return the Angle

Since the range of the arcsine function is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), the corresponding angle can only be \( \frac{\pi}{4} \), our final answer.