Problem
Evaluate: \( \sin\left(\tan^{-1}\left(\frac{24}{7}\right)\right) \)
Solution
Let \( \theta = \tan^{-1}\left(\frac{24}{7}\right) \)
Then:
\[ \tan \theta = \frac{24}{7} \]
Construct a right triangle:
- Opposite = 24
- Adjacent = 7
Hypotenuse:
\[ \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]
Now,
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{24}{25} \]
Therefore:
\[ \sin\left(\tan^{-1}\left(\frac{24}{7}\right)\right) = \frac{24}{25} \]
Final Answer
\[ \boxed{\frac{24}{25}} \]
Explanation
We convert the inverse tangent into a right triangle ratio and then use the sine definition.