Question

question 13
find the reference angle, the quadrant of the terminal side, and the sine and cosine of the angle.
\\( \frac { 5 \pi } { 3 } \\)
reference angle:
quadrant:
\\( \sin \left( \frac { 5 \pi } { 3 } \
ight) = \\)
\\( \cos \left( \frac { 5 \pi } { 3 } \
ight) = \\)
question help: message instructor

Explanation:

Step1: Determine the quadrant

Since \(2\pi=\frac{6\pi}{3}\) and \(\frac{3\pi}{2}=\frac{9\pi}{6}=\frac{4.5\pi}{3}\), and \(\frac{3\pi}{2}<\frac{5\pi}{3}<2\pi\), the terminal side of the angle \(\frac{5\pi}{3}\) is in Quadrant IV.

Step2: Calculate the reference angle

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in Quadrant IV is \(\theta_{r} = 2\pi-\theta\). So, \(\theta_{r}=2\pi - \frac{5\pi}{3}=\frac{6\pi - 5\pi}{3}=\frac{\pi}{3}\)

Step3: Find \(\sin(\frac{5\pi}{3})\)

We know that \(\sin\theta=-\sin\theta_{r}\) for \(\theta\) in Quadrant IV. Since \(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\), then \(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\)

Step4: Find \(\cos(\frac{5\pi}{3})\)

We know that \(\cos\theta=\cos\theta_{r}\) for \(\theta\) in Quadrant IV. Since \(\cos(\frac{\pi}{3})=\frac{1}{2}\), then \(\cos(\frac{5\pi}{3})=\frac{1}{2}\)

Answer:

Reference angle: \(\frac{\pi}{3}\)
Quadrant: IV
\(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\)
\(\cos(\frac{5\pi}{3})=\frac{1}{2}\)