To find the exact value of [tex]\(\arccos \left(-\frac{1}{2}\right)\)[/tex], let's follow the steps:
1. Understand the Problem:
- The [tex]\(\arccos\)[/tex] function is the inverse of the cosine function and it returns an angle whose cosine is the given value.
- We need the angle [tex]\(\theta\)[/tex] such that [tex]\(\cos \theta = -\frac{1}{2}\)[/tex].
2. Recall the Range of arcsine Function:
- The [tex]\(\arccos\)[/tex] function outputs angles in the range [tex]\([0, \pi]\)[/tex].
3. Determine the Angle:
- We look for an angle [tex]\(\theta\)[/tex] in the range [tex]\([0, \pi]\)[/tex] for which the cosine is [tex]\(-\frac{1}{2}\)[/tex].
- The cosine of [tex]\(\frac{2\pi}{3}\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
Verification by Unit Circle:
- On the unit circle, [tex]\(\cos \left(\frac{2\pi}{3}\right) = -\frac{1}{2}\)[/tex].
- This means [tex]\(\theta = \frac{2\pi}{3}\)[/tex] is the angle in the range [tex]\([0, \pi]\)[/tex] that satisfies the condition.
So, based on our analysis, we conclude that:
The exact value of [tex]\(\arccos \left(-\frac{1}{2}\right)\)[/tex] is [tex]\(\boxed{\frac{2 \pi}{3}}\)[/tex].
