Problem
Evaluate: \( \sin\left(\cos^{-1}\left(\frac{5}{13}\right)\right) \)
Solution
Let \( \theta = \cos^{-1}\left(\frac{5}{13}\right) \)
Then:
\[ \cos \theta = \frac{5}{13} \]
Using identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
\[ \sin \theta = \sqrt{1 – \cos^2 \theta} \]
\[ \sin \theta = \sqrt{1 – \left(\frac{5}{13}\right)^2} = \sqrt{1 – \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \]
Therefore:
\[ \sin\left(\cos^{-1}\left(\frac{5}{13}\right)\right) = \frac{12}{13} \]
Final Answer
\[ \boxed{\frac{12}{13}} \]
Explanation
We use a right triangle where adjacent = 5 and hypotenuse = 13. Then opposite = 12 using Pythagoras theorem. So sin θ = opposite/hypotenuse = 12/13.