Question

without using a calculator, compute the sine and cosine of \\( \frac { 5 \pi } { 3 } \\) by using the reference angle.
what is the reference angle? radians.
in what quadrant is this angle? (answer 1, 2, 3, or 4)
\\( \sin \left( \frac { 5 \pi } { 3 } \
ight) = \\)
\\( \cos \left( \frac { 5 \pi } { 3 } \
ight) = \\)
(type sqrt(2) for \\( \sqrt { 2 } \\) and sqrt(3) for \\( \sqrt { 3 } \\).)
question help: video 1 video 2

Explanation:

Step1: Determine the quadrant

Since \(2\pi=\frac{6\pi}{3}\) and \(\frac{3\pi}{2}=\frac{4.5\pi}{3}\), and \(\frac{3\pi}{2}<\frac{5\pi}{3}<2\pi\), the angle \(\frac{5\pi}{3}\) is in the fourth quadrant.

Step2: Find the reference angle

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the fourth quadrant 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: Calculate \(\sin(\frac{5\pi}{3})\)

In the fourth quadrant, \(\sin\theta<0\). Using the identity \(\sin\theta=-\sin\theta_{r}\), since \(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\), then \(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\)

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

In the fourth quadrant, \(\cos\theta>0\). Using the identity \(\cos\theta=\cos\theta_{r}\), since \(\cos(\frac{\pi}{3})=\frac{1}{2}\), then \(\cos(\frac{5\pi}{3})=\frac{1}{2}\)

Answer:

Reference angle: \(\frac{\pi}{3}\) radians.
Quadrant: \(4\)
\(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\)
\(\cos(\frac{5\pi}{3})=\frac{1}{2}\)