Question
Evaluate:
\[ \sin(\tan^{-1}\left(\frac{3}{4}\right)) \]
Solution
Let
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]
Then,
\[ \tan \theta = \frac{3}{4} = \frac{\text{Opposite}}{\text{Adjacent}} \]
Take a right triangle:
- Opposite = 3
- Adjacent = 4
Hypotenuse:
\[ \sqrt{3^2 + 4^2} = 5 \]
Now,
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} \]
Final Answer:
\[ \boxed{\frac{3}{5}} \]
Key Concept
Convert inverse trigonometric expressions into triangle form for quick evaluation.