If `sin theta=(4)/(5)`, find the value of `sin2theta`:

Options

A. `24//25`
B. `16//25`
C. `9//20`
D. none of these

Answer

To find the value of \( \sin 2\theta \) given that \( \sin \theta = \frac{4}{5} \), we can follow these steps:

### Step 1: Use the double angle formula for sine
The formula for \( \sin 2\theta \) is:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]

### Step 2: Find \( \cos \theta \)
We know that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Substituting \( \sin \theta = \frac{4}{5} \):
\[ \cos^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
Calculating \( \left(\frac{4}{5}\right)^2 \):
\[ \cos^2 \theta + \frac{16}{25} = 1 \]
Now, subtract \( \frac{16}{25} \) from both sides:
\[ \cos^2 \theta = 1 - \frac{16}{25} \]
To combine the fractions, convert 1 to have a denominator of 25:
\[ \cos^2 \theta = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root gives:
\[ \cos \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 3: Substitute values into the double angle formula
Now we can substitute \( \sin \theta \) and \( \cos \theta \) back into the double angle formula:
\[ \sin 2\theta = 2 \left(\frac{4}{5}\right) \left(\frac{3}{5}\right) \]
Calculating this gives:
\[ \sin 2\theta = 2 \cdot \frac{4 \cdot 3}{5 \cdot 5} = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Final Answer
Thus, the value of \( \sin 2\theta \) is:
\[ \sin 2\theta = \frac{24}{25} \]

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