To evaluate the expression \( \arccos \left(-\frac{1}{2}\right) \), let's follow a logical step-by-step process:
1. Understanding Arccosine: The arccosine function, \( \arccos(x) \), is the inverse of the cosine function. It returns the angle whose cosine is \( x \). The range of the \( \arccos \) function is from 0 to \( \pi \) radians.
2. Identifying the Special Value: We are given \(\arccos \left(-\frac{1}{2}\right)\). We need to determine an angle within the range \( [0, \pi] \) such that its cosine is \(-\frac{1}{2}\).
3. Cosine Properties: Recall the unit circle and the common angles:
- \( \cos(\pi/3) = 1/2 \)
- \( \cos(2\pi/3) = -1/2 \)
4. Selecting the Correct Angle: Since we require \( \cos(\theta) = -1/2 \), we look for the angle \( \theta \) in the range \( [0, \pi] \):
- \( \cos(2\pi/3) = -1/2 \)
Therefore, the angle whose cosine is \(-1/2\) within the given range is \( 2\pi/3 \).
5. Simplified Fraction: The angle \( 2\pi/3 \) doesn't need further simplification. It's already presented as a simplified fraction in terms of \(\pi\).
Hence, the simplified fraction of the angle in radians is:
[tex]\[
\arccos \left(-\frac{1}{2}\right) = \frac{2\pi}{3}
\][/tex]
