Question

Find the principal value of \(\sin^{-1}\left(\dfrac{1}{\sqrt2}\right)\).

Hint:

Recall sin(pi/4) = 1/sqrt2, and pi/4 lies in the principal range of sin^-1.

Answer 1

Step 1: Set up the equation:
Let \(\theta=\sin^{-1}\left(\dfrac1{\sqrt2}\right)\), so \(\sin\theta=\dfrac1{\sqrt2}\), with \(\theta\in\left[-\dfrac\pi2,\dfrac\pi2\right]\) (the principal range).

Step 2: Identify the angle:
\(\sin\left(\dfrac\pi4\right)=\dfrac1{\sqrt2}\), and \(\dfrac\pi4\) lies in the principal range.

Final Answer:
\[ \boxed{\theta=\dfrac\pi4} \]