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 for sine

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 \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).
Given the opposite side \(= 7\), adjacent side \(= 24\), and hypotenuse \(= 25\).
So, \(\sin\theta=\frac{7}{25}\) and \(\cos\theta=\frac{24}{25}\).

Step3: Substitute into the double - angle formula

\(\sin(2\theta)=2\times\frac{7}{25}\times\frac{24}{25}\)
\(=\frac{2\times7\times24}{25\times25}\)
\(=\frac{336}{625}\)

Answer:

\(\frac{336}{625}\)