Question

find the reference angle, the quadrant of the terminal side, and the sine and cosine of the angle.
\\( \frac { 5 \pi } { 6 } \\)
reference angle:
quadrant:
\\( \sin \left( \frac { 5 \pi } { 6 } \
ight) = \\)
\\( \cos \left( \frac { 5 \pi } { 6 } \
ight) = \\)
question help: message instructor
submit question

Explanation:

Step1: Determine the quadrant

Since \(\frac{\pi}{2}<\frac{5\pi}{6}<\pi\), the angle \(\frac{5\pi}{6}\) is in the second quadrant.

Step2: Calculate the reference angle

The formula for the reference angle \(\theta'\) of an angle \(\theta\) in the second quadrant is \(\theta'=\pi - \theta\).
So, \(\theta'=\pi-\frac{5\pi}{6}=\frac{\pi}{6}\)

Step3: Find \(\sin(\frac{5\pi}{6})\)

Using the identity \(\sin(\theta)=\sin(\pi - \theta')\), and since \(\sin(\frac{\pi}{6})=\frac{1}{2}\), then \(\sin(\frac{5\pi}{6})=\frac{1}{2}\)

Step4: Find \(\cos(\frac{5\pi}{6})\)

Using the identity \(\cos(\theta)=-\cos(\theta')\) (because in the second quadrant, cosine is negative), and since \(\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\), then \(\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}\)

Answer:

Reference angle: \(\frac{\pi}{6}\)
Quadrant: II
\(\sin(\frac{5\pi}{6})=\frac{1}{2}\)
\(\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}\)