Question
Find the value of:
\[ \cos^{-1}(\cos \tfrac{5\pi}{4}) \]
Solution
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Now,
\[ \frac{5\pi}{4} = \pi + \frac{\pi}{4} \]
We use identity:
\[ \cos(\pi + \theta) = -\cos \theta \]
So,
\[ \cos \tfrac{5\pi}{4} = -\frac{1}{\sqrt{2}} \]
Now evaluate:
\[ \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) \]
We know:
\[ \cos \tfrac{3\pi}{4} = -\frac{1}{\sqrt{2}} \]
And \( \tfrac{3\pi}{4} \in [0, \pi] \),
\[ \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = \tfrac{3\pi}{4} \]
Final Answer:
\[ \boxed{\tfrac{3\pi}{4}} \]
Key Concept
Always bring the angle into the principal range \([0, \pi]\) for inverse cosine.