To determine which trigonometric expressions equal [tex]\(-\frac{1}{2}\)[/tex], we need to evaluate each expression individually:
### 1. [tex]\(\cos(120^\circ)\)[/tex]
The cosine of [tex]\(120^\circ\)[/tex] can be evaluated using the unit circle:
[tex]\[
\cos(120^\circ) = -0.5
\][/tex]
### 2. [tex]\(\sin\left(\frac{7\pi}{6}\right)\)[/tex]
To find [tex]\(\sin\left(\frac{7\pi}{6}\right)\)[/tex], we recognize that [tex]\(\frac{7\pi}{6}\)[/tex] is in the third quadrant, where sine is negative:
[tex]\[
\sin\left(\frac{7\pi}{6}\right) = -0.5
\][/tex]
### 3. [tex]\(\sin(-120^\circ)\)[/tex]
For [tex]\(\sin(-120^\circ)\)[/tex], we consider the reference angle in the fourth quadrant where sine is negative:
[tex]\[
\sin(-120^\circ) \approx -0.866
\][/tex]
Thus, it does not equal [tex]\(-\frac{1}{2}\)[/tex].
### 4. [tex]\(\cos\left(\frac{7\pi}{6}\right)\)[/tex]
In the third quadrant, cosine is also negative:
[tex]\[
\cos\left(\frac{7\pi}{6}\right) \approx -0.866
\][/tex]
Hence, this value is not equal to [tex]\(-\frac{1}{2}\)[/tex].
### 5. [tex]\(\cos\left(-\frac{10\pi}{3}\right)\)[/tex]
First, simplify the angle [tex]\(-\frac{10\pi}{3}\)[/tex]. We recognize that rotating by [tex]\(2\pi\)[/tex] does not change the trigonometric ratios. After simplifying, the equivalent angle is:
[tex]\[
-\frac{10\pi}{3} \equiv -\frac{10\pi}{3} + 4\pi = -\frac{10\pi}{3} + \frac{12\pi}{3} = \frac{2\pi}{3}
\][/tex]
Thus:
[tex]\[
\cos\left(\frac{2\pi}{3}\right) = -0.5
\][/tex]
### Summary
The expressions that equals [tex]\(-\frac{1}{2}\)[/tex] are:
- [tex]\(\cos(120^\circ)\)[/tex]
- [tex]\(\sin\left(\frac{7\pi}{6}\right)\)[/tex]
- [tex]\(\cos\left(-\frac{10\pi}{3}\right)\)[/tex]
Thus, the correct answers are:
- [tex]\(\cos(120^\circ)\)[/tex]
- [tex]\(\sin\left(\frac{7\pi}{6}\right)\)[/tex]
- [tex]\(\cos\left(-\frac{10\pi}{3}\right)\)[/tex]
