Problem
Evaluate: \( \sec\left(\sin^{-1}\left(\frac{12}{13}\right)\right) \)
Solution
Let \( \theta = \sin^{-1}\left(\frac{12}{13}\right) \)
Then:
\[ \sin \theta = \frac{12}{13} \]
Construct a right triangle:
- Opposite = 12
- Hypotenuse = 13
Adjacent:
\[ \sqrt{13^2 – 12^2} = \sqrt{169 – 144} = \sqrt{25} = 5 \]
Now,
\[ \cos \theta = \frac{5}{13} \]
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{13}{5} \]
Therefore:
\[ \sec\left(\sin^{-1}\left(\frac{12}{13}\right)\right) = \frac{13}{5} \]
Final Answer
\[ \boxed{\frac{13}{5}} \]
Explanation
Convert sine into a right triangle, find cosine, then take reciprocal to get secant.