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

### Step 1: Convert the decimal to a fraction
We start with \( \tan^{-1}(0.75) \). We can express \( 0.75 \) as a fraction:
\[ 0.75 = \frac{75}{100} = \frac{3}{4} \]
Thus, we have:
\[ \tan^{-1}(0.75) = \tan^{-1}\left(\frac{3}{4}\right) \]

### Step 2: Use the double angle formula for sine
We know that:
\[ \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) \]
In our case, let \( \theta = \tan^{-1}\left(\frac{3}{4}\right) \). Therefore:
\[ \sin[2 \tan^{-1}\left(\frac{3}{4}\right)] = 2 \sin\left(\tan^{-1}\left(\frac{3}{4}\right)\right) \cos\left(\tan^{-1}\left(\frac{3}{4}\right)\right) \]

### Step 3: Find \( \sin(\theta) \) and \( \cos(\theta) \)
From the definition of tangent, we have:
\[ \tan(\theta) = \frac{3}{4} \]
This means that in a right triangle, the opposite side is 3 and the adjacent side is 4. We can find the hypotenuse using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
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 back into the sine double angle formula
Now substituting these values back into our formula:
\[ \sin[2 \tan^{-1}\left(\frac{3}{4}\right)] = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} \]
Calculating this gives:
\[ = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Step 5: Final answer
Thus, the value of \( \sin[2 \tan^{-1}(0.75)] \) is:
\[ \frac{24}{25} \]