To find the value of [tex]\(\cos \left( \frac{4 \pi}{3} \right)\)[/tex], consider the standard position of the angle on the unit circle.
1. Determine the Reference Angle:
[tex]\[
\frac{4\pi}{3} \text{ radians}
\][/tex]
- Since [tex]\(\pi\)[/tex] radians is equivalent to 180 degrees, [tex]\(\frac{4\pi}{3} \approx 240\)[/tex] degrees.
2. Identify the Quadrant:
- [tex]\(\frac{4\pi}{3}\)[/tex] radians (or 240 degrees) falls in the third quadrant of the unit circle.
3. Evaluate the Cosine in the Third Quadrant:
- In the third quadrant, cosine values are negative.
4. Reference Angle for the Third Quadrant:
- Calculate the reference angle: [tex]\(\frac{4\pi}{3} - \pi = \frac{\pi}{3}\)[/tex].
5. Cosine of the Reference Angle:
- The cosine of [tex]\(\frac{\pi}{3}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
6. Applying the Sign:
- Since we are in the third quadrant where cosine is negative:
[tex]\[
\cos \left( \frac{4 \pi}{3} \right) = - \cos \left(\frac{\pi}{3}\right) = -\frac{1}{2}
\][/tex]
Therefore, the value of [tex]\(\cos \left(\frac{4 \pi}{3}\right)\)[/tex] is:
[tex]\[
-\frac{1}{2}
\][/tex]
So, the correct answer is:
[tex]\[
-\frac{1}{2}
\][/tex]
