Question:
Find the principal value of:
\[ \sin^{-1}\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right) \]
Solution:
Step 1: Use identity
Recall:
\[ \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} – \sqrt{2}}{4} \]
Now simplify given expression:
\[ \frac{\sqrt{3}-1}{2\sqrt{2}} = \frac{\sqrt{6} – \sqrt{2}}{4} \]
Step 2: Substitute
\[ \sin^{-1}\left(\frac{\sqrt{6} – \sqrt{2}}{4}\right) \]
Step 3: Recognize value
\[ = \frac{\pi}{12} \]
Step 4: Check principal range
\[ \frac{\pi}{12} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
—Final Answer:
\[ \boxed{\frac{\pi}{12}} \]