Question

without using a calculator, compute the sine and cosine of 210° by using the reference angle. what is the reference angle? degrees. in what quadrant is this angle? (answer 1, 2, 3, or 4) sin(210°) = cos(210°) = (type sqrt(2) for √2 and sqrt(3) for √3.) question help: video

Explanation:

Step1: Find the reference angle

For an angle \(\theta = 210^{\circ}\), since \(180^{\circ}<210^{\circ}<270^{\circ}\), the reference angle \(\theta_{r}=\theta - 180^{\circ}\).
\(\theta_{r}=210^{\circ}-180^{\circ}=30^{\circ}\)

Step2: Determine the quadrant

Since \(180^{\circ}<210^{\circ}<270^{\circ}\), the angle \(210^{\circ}\) is in the third quadrant.

Step3: Calculate \(\sin(210^{\circ})\)

In the third quadrant, \(\sin\theta<0\). Using the identity \(\sin\theta=-\sin\theta_{r}\), with \(\theta_{r} = 30^{\circ}\) and \(\sin30^{\circ}=\frac{1}{2}\), we have \(\sin(210^{\circ})=-\frac{1}{2}\)

Step4: Calculate \(\cos(210^{\circ})\)

In the third quadrant, \(\cos\theta<0\). Using the identity \(\cos\theta=-\cos\theta_{r}\), with \(\theta_{r}=30^{\circ}\) and \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\cos(210^{\circ})=-\frac{\sqrt{3}}{2}\)

Answer:

Reference angle: \(30\) degrees.
Quadrant: \(3\)
\(\sin(210^{\circ})=-\frac{1}{2}\)
\(\cos(210^{\circ})=-\frac{\sqrt{3}}{2}\)