Question

use the unit circle to find the value of \\( \sin \frac{5\pi}{3} \\) and even or odd trigonometric functions to find the value of \\( \sin \left( -\frac{5\pi}{3} \
ight) \\). select the correct choice below and fill in any answer boxes within your choice. \\( \boldsymbol{\text{a.}} \\) \\( \sin \frac{5\pi}{3} = \square \\) (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) \\( \boldsymbol{\text{b.}} \\) the expression is undefined. unit circle diagram with coordinates: \\( (0,1) \\), \\( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \
ight) \\), \\( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \
ight) \\), \\( (1,0) \\), \\( \left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \
ight) \\), \\( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \
ight) \\), \\( (0,-1) \\), \\( \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \
ight) \\), \\( \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \
ight) \\), \\( (-1,0) \\), \\( \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \
ight) \\), \\( \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \
ight) \\)

Explanation:

Step1: Recall the unit circle definition

On the unit circle, for an angle \(\theta\), the coordinates of the point are \((\cos\theta, \sin\theta)\). So we need to find the angle \(\frac{5\pi}{3}\) on the unit circle and identify its \(y\)-coordinate (since \(\sin\theta\) is the \(y\)-coordinate).

Step2: Locate \(\frac{5\pi}{3}\) on the unit circle

The angle \(\frac{5\pi}{3}\) is in the fourth quadrant. We know that \(\frac{5\pi}{3}=2\pi - \frac{\pi}{3}\), so it has a reference angle of \(\frac{\pi}{3}\). The coordinates for an angle with reference angle \(\frac{\pi}{3}\) in the fourth quadrant will have a positive \(x\)-coordinate and negative \(y\)-coordinate. The coordinates for \(\frac{\pi}{3}\) are \((\frac{1}{2}, \frac{\sqrt{3}}{2})\), so for \(\frac{5\pi}{3}\), the coordinates are \((\frac{1}{2}, -\frac{\sqrt{3}}{2})\). Since \(\sin\theta\) is the \(y\)-coordinate, \(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\).

Answer:

\(-\frac{\sqrt{3}}{2}\)