Question
Find the value of:
\[ \sec^{-1}\left(\frac{1}{2}\right) \]
Solution
We know:
\[ \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \]
So,
\[ \sec^{-1}\left(\frac{1}{2}\right) = \cos^{-1}(2) \]
But cosine function satisfies:
\[ -1 \le \cos \theta \le 1 \]
Since \( 2 \) is outside this range,
\[ \cos^{-1}(2) \text{ is not defined in real numbers} \]
Final Answer:
\[ \boxed{\text{Not defined (no real value)}} \]
Key Concept
The domain of \( \sec^{-1}x \) is \( |x| \ge 1 \). Since \( \frac{1}{2} \in (-1,1) \), it has no real value.