If `tan theta = 3//4 and theta` is acute, then what is the value of `sintheta` ?

Options

A. `-(3)/5`
B. `3/5`
C. `4/5`
D. `-(4)/5`

Answer

To find the value of \( \sin \theta \) given that \( \tan \theta = \frac{3}{4} \) and \( \theta \) is acute, we can follow these steps:

### Step 1: Understand the relationship of tangent
We know that:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \]
From the given information, we have:
\[ \tan \theta = \frac{3}{4} \]
This means that the opposite side (perpendicular) is 3 and the adjacent side (base) is 4.

### Step 2: Draw a right triangle
We can visualize this with a right triangle where:
- The opposite side (perpendicular) is 3.
- The adjacent side (base) is 4.
- The angle opposite the side of length 3 is \( \theta \).

### Step 3: Use the Pythagorean theorem to find the hypotenuse
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ \text{Hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
Taking the square root gives:
\[ \text{Hypotenuse} = \sqrt{25} = 5 \]

### Step 4: Calculate \( \sin \theta \)
The sine of an angle in a right triangle is given by:
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Substituting the values we have:
\[ \sin \theta = \frac{3}{5} \]

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

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