Problem
Evaluate: \( \csc\left(\cos^{-1}\left(\frac{3}{5}\right)\right) \)
Solution
Let \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \)
Then:
\[ \cos \theta = \frac{3}{5} \]
Using identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
\[ \sin \theta = \sqrt{1 – \cos^2 \theta} = \sqrt{1 – \left(\frac{3}{5}\right)^2} = \sqrt{1 – \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
Now,
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]
Therefore:
\[ \csc\left(\cos^{-1}\left(\frac{3}{5}\right)\right) = \frac{5}{4} \]
Final Answer
\[ \boxed{\frac{5}{4}} \]
Explanation
Convert cosine to sine using identity, then take reciprocal to get cosecant.