Answer :
To determine which expression is equivalent to [tex]\(\cos 120^\circ\)[/tex], we need to carefully analyze the properties and behavior of the cosine function.
1. Understanding the Cosine Function:
The cosine function is periodic with a period of [tex]\(360^\circ\)[/tex], which means [tex]\(\cos(\theta) = \cos(\theta + 360^\circ k)\)[/tex] where [tex]\(k\)[/tex] is any integer.
2. Cosine of Supplementary Angles:
The cosine function exhibits symmetry with respect to [tex]\(180^\circ\)[/tex]. In other words:
[tex]\[ \cos(180^\circ + \theta) = -\cos(\theta) \][/tex]
3. Finding [tex]\(\cos 120^\circ\)[/tex]:
Specifically for [tex]\(\theta = 120^\circ\)[/tex], we observe the following:
[tex]\[ 120^\circ = 180^\circ - 60^\circ \][/tex]
By using the identity [tex]\(\cos(180^\circ - \theta) = -\cos(\theta)\)[/tex], it follows that:
[tex]\[ \cos 120^\circ = -\cos 60^\circ \][/tex]
However, this directly does not help us find an equivalent positive angle.
4. Better Equivalent Representation:
We know from the periodic property that adding [tex]\(360^\circ\)[/tex] does not change the function, so we use the property [tex]\(\cos(180 + \theta) = -\cos(\theta)\)[/tex]:
[tex]\[ 120^\circ + 120^\circ = 240^\circ \][/tex]
This gives us:
[tex]\[ \cos 240^\circ = \cos(180^\circ + 60^\circ) = -\cos 60^\circ \][/tex]
Hence:
[tex]\[ \cos 240^\circ = -\cos 60^\circ = \cos 120^\circ \][/tex]
Through these steps, it’s evident that the expression [tex]\(\cos 240^\circ\)[/tex] is equivalent to [tex]\(\cos 120^\circ\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{\cos 240^\circ} \][/tex]
1. Understanding the Cosine Function:
The cosine function is periodic with a period of [tex]\(360^\circ\)[/tex], which means [tex]\(\cos(\theta) = \cos(\theta + 360^\circ k)\)[/tex] where [tex]\(k\)[/tex] is any integer.
2. Cosine of Supplementary Angles:
The cosine function exhibits symmetry with respect to [tex]\(180^\circ\)[/tex]. In other words:
[tex]\[ \cos(180^\circ + \theta) = -\cos(\theta) \][/tex]
3. Finding [tex]\(\cos 120^\circ\)[/tex]:
Specifically for [tex]\(\theta = 120^\circ\)[/tex], we observe the following:
[tex]\[ 120^\circ = 180^\circ - 60^\circ \][/tex]
By using the identity [tex]\(\cos(180^\circ - \theta) = -\cos(\theta)\)[/tex], it follows that:
[tex]\[ \cos 120^\circ = -\cos 60^\circ \][/tex]
However, this directly does not help us find an equivalent positive angle.
4. Better Equivalent Representation:
We know from the periodic property that adding [tex]\(360^\circ\)[/tex] does not change the function, so we use the property [tex]\(\cos(180 + \theta) = -\cos(\theta)\)[/tex]:
[tex]\[ 120^\circ + 120^\circ = 240^\circ \][/tex]
This gives us:
[tex]\[ \cos 240^\circ = \cos(180^\circ + 60^\circ) = -\cos 60^\circ \][/tex]
Hence:
[tex]\[ \cos 240^\circ = -\cos 60^\circ = \cos 120^\circ \][/tex]
Through these steps, it’s evident that the expression [tex]\(\cos 240^\circ\)[/tex] is equivalent to [tex]\(\cos 120^\circ\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{\cos 240^\circ} \][/tex]
