We need to simplify the expression \( \sin(\tan^{-1}(3/4)) \).

### Step-by-Step Solution:
1. Let \( \theta = \tan^{-1}(3/4) \). This means that \( \tan(\theta) = 3/4 \).
   
2. From the definition of the tangent function, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Therefore, if \( \tan(\theta) = 3/4 \), we can construct a right triangle where:
   - Opposite side = 3
   - Adjacent side = 4

3. Now, we can use the Pythagorean theorem to find the hypotenuse:
   \[
   \text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   \]
   So the hypotenuse is 5.

4. We need to find \( \sin(\theta) \). From the definition of sine:
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}
   \]

Therefore, \( \sin(\tan^{-1}(3/4)) = \frac{3}{5} \).

### Final Answer:
\[
\sin(\tan^{-1}(3/4)) = \frac{3}{5}
\]

Would you like more details on this problem, or do you have any other questions?

Here are 5 related questions:
1. How do you simplify \( \cos(\tan^{-1}(3/4)) \)?
2. What is \( \tan(\sin^{-1}(4/5)) \)?
3. Can you find \( \sin(\cot^{-1}(4/3)) \)?
4. What is \( \cos(\sin^{-1}(3/5)) \)?
5. How do you simplify \( \tan(\sec^{-1}(5/4)) \)?

**Tip**: When simplifying inverse trigonometric functions, it's often helpful to visualize them using right triangles.

This page simplifies the expression sin(tan^-1(3/4)) using trigonometric concepts and the Pythagorean theorem. Follow the step-by-step guide to find the sine of the angle whose tangent is 3/4. Perfect for students in Grades 10-12.

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