Question

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

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 \(\sin(2\theta)=2\sin\theta\cos\theta\).

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Answer:

\(\frac{24}{25}\)