Problem
Evaluate: \( \cos\left(\sin^{-1}\left(\frac{-7}{25}\right)\right) \)
Solution
Let \( \theta = \sin^{-1}\left(\frac{-7}{25}\right) \)
Then:
\[ \sin \theta = \frac{-7}{25} = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
- Perpendicular = -7
- Hypotenuse = 25
Base:
\[ \sqrt{25^2 – 7^2} = \sqrt{625 – 49} = \sqrt{576} = 24 \]
Now,
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{24}{25} \]
Therefore:
\[ \cos\left(\sin^{-1}\left(\frac{-7}{25}\right)\right) = \frac{24}{25} \]
Final Answer
\[ \boxed{\frac{24}{25}} \]
Explanation
Using identity: cos(sin⁻¹x) = √(1 − x²). Cosine is positive in the principal range of sin⁻¹x.