Problem
Evaluate: \( \cos\left(\tan^{-1}\left(\frac{24}{7}\right)\right) \)
Solution
Let \( \theta = \tan^{-1}\left(\frac{24}{7}\right) \)
Then:
\[ \tan \theta = \frac{24}{7} = \frac{\text{Perpendicular}}{\text{Base}} \]
- Perpendicular = 24
- Base = 7
Hypotenuse:
\[ \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]
Now, using:
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} \]
\[ \cos \theta = \frac{7}{25} \]
Therefore:
\[ \cos\left(\tan^{-1}\left(\frac{24}{7}\right)\right) = \frac{7}{25} \]
Final Answer
\[ \boxed{\frac{7}{25}} \]
Explanation
Using right triangle definitions: tan = perpendicular/base and cos = base/hypotenuse.