Question

use reference angles to find the exact value of the following expression. sin(4π/3) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. sin(4π/3)= (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator.) b. the answer is undefined.

Explanation:

Step1: Determine the quadrant

The angle $\frac{4\pi}{3}$ is in the third - quadrant since $\pi<\frac{4\pi}{3}<\frac{3\pi}{2}$.

Step2: Find the reference angle

The reference angle $\theta_{r}$ for an angle $\theta=\frac{4\pi}{3}$ in the third - quadrant is $\theta_{r}=\frac{4\pi}{3}-\pi=\frac{\pi}{3}$.

Step3: Determine the sign of sine in the third - quadrant

In the third - quadrant, $\sin\theta<0$.

Step4: Evaluate the sine of the reference angle

We know that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$. Since $\sin\theta$ is negative in the third - quadrant, $\sin\frac{4\pi}{3}=-\sin\frac{\pi}{3}$.
So, $\sin\frac{4\pi}{3}=-\frac{\sqrt{3}}{2}$.

Answer:

A. $\sin\frac{4\pi}{3}=-\frac{\sqrt{3}}{2}$