In a right-angled triangle, if $\sin \theta =\frac{3}{5}$, what is the value of $2\sin \theta \cos \theta$?

The correct answer is $\frac{24}{25}$.

Step 1: Understand the given trigonometric ratio.

In a right-angled triangle, the sine function of an angle $\theta$ is defined as the ratio of the length of the opposite side to the hypotenuse:

$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{3}{5}$

Therefore, we can represent the opposite side as 3 units and the hypotenuse as 5 units.

Step 2: Find the length of the adjacent side using Pythagoras' theorem.

According to Pythagoras' theorem in a right-angled triangle:

${\text{Hypotenuse}}^2={\text{Opposite}}^2+{\text{Adjacent}}^2$

Substituting the known values into the formula:

$5^2=3^2+{\text{Adjacent}}^2$

$25=9+{\text{Adjacent}}^2$

${\text{Adjacent}}^2=25-9=16$

$\text{Adjacent}=\sqrt{16}=4$

Step 3: Determine the value of $\cos \theta$.

The cosine function is defined as the ratio of the adjacent side to the hypotenuse:

$\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{4}{5}$

Step 4: Calculate the value of $2\sin \theta \cos \theta$.

Now, substitute the values of $\sin \theta$ and $\cos \theta$ into the given expression:

$2\sin \theta \cos \theta =2\times \frac{3}{5}\times \frac{4}{5}$

$2\sin \theta \cos \theta =\frac{2\times 3\times 4}{5\times 5}=\frac{24}{25}$

Hence, the value of $2\sin \theta \cos \theta$ is $\frac{24}{25}$.