Use the figure below to determine the exact value of the given function.
$\sin 2 \theta$

A) $\sin 2 \theta = \frac { 5 } { 13 }$
B) $\sin 2 \theta = \frac { 7 } { 13 }$
C) $\sin 2 \theta = \frac { 12 } { 5 }$
D) $\sin 2 \theta = \frac { 9 } { 13 }$
E) $\sin 2 \theta = \frac { 12 } { 13 }$

Question

Use the figure below to determine the exact value of the given function.
$\sin 2 \theta$

A) $\sin 2 \theta = \frac { 5 } { 13 }$ 
B) $\sin 2 \theta = \frac { 7 } { 13 }$ 
C) $\sin 2 \theta = \frac { 12 } { 5 }$ 
D) $\sin 2 \theta = \frac { 9 } { 13 }$ 
E) $\sin 2 \theta = \frac { 12 } { 13 }$

Quizplus · Accepted Answer

We can use the Pythagorean theorem to find the length of the opposite side of the triangle: 
$$\sqrt{13^2-5^2}=\sqrt{144}=12.$$ 
Thus, we have $\sin	heta=\frac{5}{13}$ and $\cos	heta=\frac{12}{13}$. Using the double angle formula for sine, we have: 
$$\sin2	heta=2\sin	heta\cos	heta=\frac{2\cdot 5\cdot 12}{13^2}=\frac{120}{169}.$$ 
Therefore, the correct choice is E.

Use the Figure Below to Determine the Exact Value of the Given