If the value of $$ \cos heta=-\frac{\sqrt{3}}{2} $$, which of the following could be true?

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If the value of $$ \cos 	heta=-\frac{\sqrt{3}}{2} $$, which of the following could be true?

UpStudy AI · Accepted Answer

To determine the possible values of $$ \theta $$ when $$ \cos \theta = -\frac{\sqrt{3}}{2} $$, we need to identify the angles where the cosine function takes this value.

The cosine function is negative in the second and third quadrants. The reference angle for $$ \cos \theta = \frac{\sqrt{3}}{2} $$ is $$ \theta = \frac{\pi}{6} $$ (or $$ 30^\circ $$). Therefore, the angles in the second and third quadrants that correspond to $$ \cos \theta = -\frac{\sqrt{3}}{2} $$ can be found as follows:

1. In the second quadrant:
   $$
   \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}
   $$

2. In the third quadrant:
   $$
   \theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}
   $$

Thus, the angles that satisfy $$ \cos \theta = -\frac{\sqrt{3}}{2} $$ are:
- $$ \theta = \frac{5\pi}{6} + 2k\pi $$ for any integer $$ k $$ (for the second quadrant)
- $$ \theta = \frac{7\pi}{6} + 2k\pi $$ for any integer $$ k $$ (for the third quadrant)

In degrees, these angles correspond to:
- $$ \theta = 150^\circ + 360^\circ k $$
- $$ \theta = 210^\circ + 360^\circ k $$

Therefore, any angle of the form $$ \frac{5\pi}{6} + 2k\pi $$ or $$ \frac{7\pi}{6} + 2k\pi $$ (or their degree equivalents) could be true for $$ \cos \theta = -\frac{\sqrt{3}}{2} $$.
The angles that satisfy $$ \cos \theta = -\frac{\sqrt{3}}{2} $$ are $$ \frac{5\pi}{6} + 2k\pi $$ and $$ \frac{7\pi}{6} + 2k\pi $$, where $$ k $$ is any integer.

If the value of \( \cos \theta=-\frac{\sqrt{3}}{2} \), which of the following could be true?

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