Find the function value using coordinates of points on the unit circle. Give an exact answer. \( \sin \frac{4 \pi}{3} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \frac{4 \pi}{3}=\square \) (Simplify your answer, including any radicals. Use integers or'fractions for any numbers in the expression.) B. The function is indefined.

To find \( \sin \frac{4 \pi}{3} \) using the unit circle, first, we locate the angle \( \frac{4 \pi}{3} \). This angle is in the third quadrant, and its reference angle is \( \pi - \frac{4 \pi}{3} = \frac{\pi}{3} \). The sine value in the third quadrant is negative, so we find: \[ \sin \frac{4 \pi}{3} = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2} \] So, the answer is: A. \( \sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2} \) The unit circle has been a vital tool for mathematicians, providing a visual representation of angles and their corresponding sine and cosine values. For instance, knowing the angle's location improves our understanding of trigonometric functions in different quadrants. When calculating sine values, one common mistake is misidentifying the quadrant. Always check the angle against the unit circle to determine the correct sign of the sine value. For instance, remember that sine is positive in the first and second quadrants and negative in the third and fourth quadrants!