Question

Which angle has a sine of
−
1
2
and a cosine of
−
√
3
2
?

A.
5
π
6

B.
7
π
6

C.
4
π
3

D.
5
π
3

Answer 1

To find the angle that has a sine of \(-\frac{1}{2}\) and a cosine of \(-\frac{\sqrt{3}}{2}\), we can analyze the sine and cosine values.

1. **Sine = -1/2**: This implies that the angle is in the 4th or 3rd quadrant, where sine is negative.

2. **Cosine = -√3/2**: This indicates the angle is in the 2nd or 3rd quadrant, where cosine is negative.

Given that sine is negative and cosine is also negative, the angle must be in the 3rd quadrant.

The reference angle corresponding to \(\sin(\theta) = \frac{1}{2}\) is \(\frac{\pi}{6}\). Thus, in the 3rd quadrant, the angle can be calculated as:

\[
\theta = \pi + \text{reference angle} = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}
\]

Now, we can check the options:

A. \( \frac{5\pi}{6} \) - Not in the 3rd quadrant
B. \( \frac{7\pi}{6} \) - This matches our finding
C. \( \frac{4\pi}{3} \) - This is \( \frac{8\pi}{6} \), which has positive sine
D. \( \frac{5\pi}{3} \) - Not in the 3rd quadrant

Thus, the angle that has a sine of \(-\frac{1}{2}\) and a cosine of \(-\frac{\sqrt{3}}{2}\) is:

**B. \( \frac{7\pi}{6} \)**.