Question

rewrite the expression in terms of the given angles reference angle; then evaluate the result. write the exact answer. do not round. sin(11π/3)

Explanation:

Step1: Find the coterminal angle

First, find a coterminal angle of $\frac{11\pi}{3}$ that lies between $0$ and $2\pi$. We know that $2\pi=\frac{6\pi}{3}$. Subtract $2\pi$ from $\frac{11\pi}{3}$ multiple - times. $\frac{11\pi}{3}-2\pi=\frac{11\pi}{3}-\frac{6\pi}{3}=\frac{5\pi}{3}$.

Step2: Determine the reference angle

The angle $\theta = \frac{5\pi}{3}$ is in the fourth - quadrant. The reference angle $\theta'$ for an angle $\theta$ in the fourth - quadrant is given by $\theta'=2\pi-\theta$. So, $\theta' = 2\pi-\frac{5\pi}{3}=\frac{6\pi}{3}-\frac{5\pi}{3}=\frac{\pi}{3}$.

Step3: Use the sign of sine in the fourth - quadrant

In the fourth - quadrant, the sine function is negative. So, $\sin(\frac{11\pi}{3})=\sin(\frac{5\pi}{3})=-\sin(\frac{\pi}{3})$.

Step4: Evaluate the sine of the reference angle

We know that $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$. So, $-\sin(\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$.

Answer:

$-\frac{\sqrt{3}}{2}$