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

Explanation:

Step1: Recall the double - angle formula

The double - angle formula for sine is (\sin(2\theta)=2\sin\theta\cos\theta).

Step2: Find (\sin\theta) and (\cos\theta)

In a right - triangle, (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}) and (\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}). For the given right - triangle with hypotenuse (c = 5), opposite side (a = 4) and adjacent side (b=3), we have (\sin\theta=\frac{4}{5}) and (\cos\theta=\frac{3}{5}).

Step3: Substitute into the double - angle formula

Substitute (\sin\theta=\frac{4}{5}) and (\cos\theta=\frac{3}{5}) into (\sin(2\theta)=2\sin\theta\cos\theta). [ \begin{align*} \sin(2\theta)&=2\times\frac{4}{5}\times\frac{3}{5}\ &=\frac{2\times4\times3}{5\times5}\ &=\frac{24}{25} \end{align*} ]

Answer:

(\frac{24}{25})