The value of `sin[2tan^(-1)(0.75)]` is

Options

A. `0.75`
B. `1.5`
C. `0.96`
D. `sin 1.5`

Explanation

To find the value of \( \sin[2 \tan^{-1}(0.75)] \), we can follow these steps:

### Step 1: Let \( \theta = \tan^{-1}(0.75) \)
This means that \( \tan(\theta) = 0.75 \).

### Step 2: Use the double angle formula for sine
We know that:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
So, we need to find \( \sin(\theta) \) and \( \cos(\theta) \).

### Step 3: Find \( \sin(\theta) \) and \( \cos(\theta) \)
Since \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} \), we can use the Pythagorean theorem to find the hypotenuse:
\[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Now we can find \( \sin(\theta) \) and \( \cos(\theta) \):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]

### Step 4: Substitute into the double angle formula
Now we can substitute \( \sin(\theta) \) and \( \cos(\theta) \) into the double angle formula:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} \]

### Step 5: Calculate the value
\[ \sin(2\theta) = 2 \cdot \frac{3 \cdot 4}{5 \cdot 5} = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Step 6: Final value
Thus, the value of \( \sin[2 \tan^{-1}(0.75)] \) is:
\[ \frac{24}{25} \approx 0.96 \]

### Summary
The final answer is \( 0.96 \).

---