Question

use the figure to find the exact value of the following trigonometric function. sin(2θ) sin(2θ)= (simplify your answer.)

Explanation:

Step1: Find $\sin\theta$ and $\cos\theta$

By the definition of sine and cosine in a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{12}{15}=\frac{4}{5}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{9}{15}=\frac{3}{5}$.

Step2: Use the double - angle formula for sine

The double - angle formula for sine is $\sin(2\theta)=2\sin\theta\cos\theta$.
Substitute $\sin\theta = \frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ into the formula:
$\sin(2\theta)=2\times\frac{4}{5}\times\frac{3}{5}$.

Step3: Calculate the value

$2\times\frac{4}{5}\times\frac{3}{5}=\frac{2\times4\times3}{5\times5}=\frac{24}{25}$.

Answer:

$\frac{24}{25}$