Question
Find the set of values of:
\[ \csc^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
Solution
We know that:
\[ \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \]
So,
\[ \csc^{-1}\left(\frac{\sqrt{3}}{2}\right) = \sin^{-1}\left(\frac{2}{\sqrt{3}}\right) \]
But,
\[ \frac{2}{\sqrt{3}} > 1 \]
And we know:
\[ -1 \le \sin \theta \le 1 \]
So,
\[ \sin^{-1}\left(\frac{2}{\sqrt{3}}\right) \text{ is not defined in real numbers} \]
Final Answer:
\[ \boxed{\text{No real value (not defined)}} \]
Key Concept
The domain of \( \csc^{-1}x \) is \( |x| \ge 1 \). Since \( \frac{\sqrt{3}}{2} < 1 \), the expression has no real value.