If `tan A = 3/4` , find the value of sin A.

Answer

To find the value of sin A given that tan A = 3/4, we can follow these steps:

### Step 1: Understand the relationship of tan, sin, and cos
The tangent of an angle A in a right triangle is defined as the ratio of the opposite side (perpendicular) to the adjacent side (base):
\[ \tan A = \frac{\text{Opposite}}{\text{Adjacent}} \]

### Step 2: Set up the triangle
Given that \( \tan A = \frac{3}{4} \), we can assign:
- Opposite side (perpendicular) = 3k
- Adjacent side (base) = 4k
where k is a positive constant.

### Step 3: Use the Pythagorean theorem to find the hypotenuse
We can find the hypotenuse (h) using the Pythagorean theorem:
\[ h^2 = (\text{Opposite})^2 + (\text{Adjacent})^2 \]
Substituting the values:
\[ h^2 = (3k)^2 + (4k)^2 \]
\[ h^2 = 9k^2 + 16k^2 \]
\[ h^2 = 25k^2 \]
Taking the square root:
\[ h = 5k \]

### Step 4: Find sin A
The sine of angle A is defined as the ratio of the opposite side to the hypotenuse:
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Substituting the values we found:
\[ \sin A = \frac{3k}{5k} \]
The k's cancel out:
\[ \sin A = \frac{3}{5} \]

### Final Answer
Thus, the value of sin A is:
\[ \sin A = \frac{3}{5} \]

---