Problem
Evaluate: \( \sin\left(\sec^{-1}\left(\frac{17}{8}\right)\right) \)
Solution
Let \( \theta = \sec^{-1}\left(\frac{17}{8}\right) \)
Then:
\[ \sec \theta = \frac{17}{8} \]
So,
\[ \cos \theta = \frac{8}{17} \]
Using identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
\[ \sin \theta = \sqrt{1 – \cos^2 \theta} = \sqrt{1 – \left(\frac{8}{17}\right)^2} = \sqrt{1 – \frac{64}{289}} = \sqrt{\frac{225}{289}} = \frac{15}{17} \]
Therefore:
\[ \sin\left(\sec^{-1}\left(\frac{17}{8}\right)\right) = \frac{15}{17} \]
Final Answer
\[ \boxed{\frac{15}{17}} \]
Explanation
Convert secant into cosine, form a right triangle, and use the Pythagorean identity to find sine.