Sure! Let's analyze the given trigonometric function step by step:
We need to find the value of [tex]\( \cos \left( \frac{5\pi}{3} \right) \)[/tex].
1. Determine the Quadrant:
[tex]\(\frac{5\pi}{3}\)[/tex] radians is an angle measured in the standard position. Since [tex]\(2\pi\)[/tex] radians represent a full circle (360 degrees), and [tex]\(\pi\)[/tex] represents half of a circle (180 degrees), we can see that:
[tex]\[
\frac{5\pi}{3} \text{ radians} = 2\pi - \frac{\pi}{3}
\][/tex]
This tells us that [tex]\(\frac{5\pi}{3}\)[/tex] radians is equivalent to [tex]\(2\pi - \frac{\pi}{3}\)[/tex], which places the angle in the fourth quadrant.
2. Reference Angle:
The reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] in the fourth quadrant is:
[tex]\[
2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}
\][/tex]
3. Cosine Value in the Fourth Quadrant:
In the fourth quadrant, the cosine function is positive. For the reference angle [tex]\(\frac{\pi}{3}\)[/tex]:
[tex]\[
\cos \left( \frac{\pi}{3} \right) = \frac{1}{2}
\][/tex]
Therefore:
[tex]\[
\cos \left( \frac{5\pi}{3} \right) = \cos \left( 2\pi - \frac{\pi}{3} \right) = \cos \left( -\frac{\pi}{3} \right) = \frac{1}{2}
\][/tex]
Given the choices:
A. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
B. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
The correct answer is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
