To find the value of [tex]\(\cos\left(\frac{5\pi}{3}\right)\)[/tex], we need to follow these steps:
1. Determine the corresponding angle in the unit circle:
- The angle [tex]\(\frac{5\pi}{3}\)[/tex] is in radians. To better understand its position, convert it to degrees by multiplying by [tex]\(\frac{180^\circ}{\pi}\)[/tex]:
[tex]\[
\frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 5 \times 60^\circ = 300^\circ
\][/tex]
2. Locate the angle on the unit circle:
- The angle [tex]\(300^\circ\)[/tex] is in the fourth quadrant of the unit circle.
3. Reference angle:
- The reference angle for [tex]\(300^\circ\)[/tex] (an angle in the fourth quadrant) can be found by subtracting it from [tex]\(360^\circ\)[/tex]:
[tex]\[
360^\circ - 300^\circ = 60^\circ
\][/tex]
- Therefore, the reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] is [tex]\(60^\circ\)[/tex] or [tex]\(\frac{\pi}{3}\)[/tex] radians.
4. Value of the cosine at the reference angle:
- We know that [tex]\(\cos(60^\circ) = \frac{1}{2}\)[/tex].
5. Considering the quadrant:
- In the fourth quadrant, the cosine value is positive because the x-coordinates of points on the unit circle are positive in this quadrant.
Therefore, [tex]\(\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}\)[/tex].
Thus, the correct option is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
This corresponds to option A.
