Question

find the reference angle, the quadrant of the terminal side, and the sine and cosine of the angle. answer exactly, do not round. 240° reference angle: quadrant: sin(240°)= cos(240°)= question help: message instructor question 10

Explanation:

Step1: Determine the quadrant

Since \(180^{\circ}<240^{\circ}<270^{\circ}\), the terminal side of \(240^{\circ}\) is in the third quadrant.

Step2: Calculate the reference angle

The formula for the reference angle \(\theta'\) of an angle \(\theta\) in the third quadrant is \(\theta'=\theta - 180^{\circ}\).
So, \(\theta'=240^{\circ}- 180^{\circ}=60^{\circ}\)

Step3: Find \(\sin(240^{\circ})\)

We know that \(\sin(240^{\circ})=-\sin(60^{\circ})\) (because sine is negative in the third quadrant)
Since \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\), then \(\sin(240^{\circ})=-\frac{\sqrt{3}}{2}\)

Step4: Find \(\cos(240^{\circ})\)

We know that \(\cos(240^{\circ})=-\cos(60^{\circ})\) (because cosine is negative in the third quadrant)
Since \(\cos(60^{\circ})=\frac{1}{2}\), then \(\cos(240^{\circ})=-\frac{1}{2}\)

Answer:

Reference angle: \(60^{\circ}\)
Quadrant: Third
\(\sin(240^{\circ})=-\frac{\sqrt{3}}{2}\)
\(\cos(240^{\circ})=-\frac{1}{2}\)