If `costheta=3/5`, 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 \( \cos \theta = \frac{3}{5} \), we can follow these steps:

### Step 1: Identify the values of \( \cos \theta \)
We know that:
\[ \cos \theta = \frac{3}{5} \]
This means that in a right triangle, the adjacent side (base) is 3 and the hypotenuse is 5.

### Step 2: Use the Pythagorean theorem to find \( \sin \theta \)
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Let \( \text{perpendicular} = y \). Then:
\[ 5^2 = y^2 + 3^2 \]
\[ 25 = y^2 + 9 \]
\[ y^2 = 25 - 9 = 16 \]
\[ y = \sqrt{16} = 4 \]
Thus, \( \sin \theta = \frac{y}{\text{hypotenuse}} = \frac{4}{5} \).

### Step 3: Use the double angle formula for sine
The formula for \( \sin 2\theta \) is:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]

### Step 4: Substitute the values of \( \sin \theta \) and \( \cos \theta \)
Now substitute the values we found:
\[ \sin 2\theta = 2 \left(\frac{4}{5}\right) \left(\frac{3}{5}\right) \]
\[ = 2 \cdot \frac{4 \cdot 3}{5 \cdot 5} \]
\[ = 2 \cdot \frac{12}{25} \]
\[ = \frac{24}{25} \]

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

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