To determine which expression is equivalent to [tex]\(\cos 120^\circ\)[/tex], let's first understand the values of the cosine function at each of the given angles.
1. [tex]\(\cos 120^\circ\)[/tex]:
[tex]\[
\cos 120^\circ = \cos (180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2}
\][/tex]
2. [tex]\(\cos 60^\circ\)[/tex]:
[tex]\[
\cos 60^\circ = \frac{1}{2}
\][/tex]
3. [tex]\(\cos 240^\circ\)[/tex]:
[tex]\[
\cos 240^\circ = \cos (180^\circ + 60^\circ) = -\cos 60^\circ = -\frac{1}{2}
\][/tex]
4. [tex]\(\cos 300^\circ\)[/tex]:
[tex]\[
\cos 300^\circ = \cos (360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2}
\][/tex]
5. [tex]\(\cos 420^\circ\)[/tex]:
[tex]\[
\cos 420^\circ = \cos (360^\circ + 60^\circ) = \cos 60^\circ = \frac{1}{2}
\][/tex]
From these calculations, we observe that:
- [tex]\(\cos 120^\circ = -\frac{1}{2}\)[/tex]
- [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex]
- [tex]\(\cos 240^\circ = -\frac{1}{2}\)[/tex]
- [tex]\(\cos 300^\circ = \frac{1}{2}\)[/tex]
- [tex]\(\cos 420^\circ = \frac{1}{2}\)[/tex]
Thus, the expression that is equivalent to [tex]\(\cos 120^\circ\)[/tex] is:
[tex]\[
\cos 240^\circ
\][/tex]
