Find each exact function value. See Example 2.
sin (4π/3)

Verified step by step guidance

1. Identify the reference angle for \(\frac{4\pi}{3}\). The reference angle is the angle in the first quadrant that corresponds to \(\frac{4\pi}{3}\). Since \(\frac{4\pi}{3}\) is in the third quadrant, subtract \(\pi\) to find the reference angle: \(\frac{4\pi}{3} - \pi = \frac{\pi}{3}\).
2. Determine the sine of the reference angle. The sine of \(\frac{\pi}{3}\) is known from the unit circle to be \(\frac{\sqrt{3}}{2}\).
3. Consider the sign of the sine function in the third quadrant. In the third quadrant, both sine and cosine are negative.
4. Apply the sign to the sine value of the reference angle. Since sine is negative in the third quadrant, \(\sin(\frac{4\pi}{3}) = -\sin(\frac{\pi}{3})\).
5. Combine the results to find the exact value: \(\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}\).