Evaluate cosine Superscript negative 1 Baseline left parenthesis cosine StartFraction 4 pi Over 3 EndFraction right parenthesis
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To evaluate \(\cos^{-1}(\cos(\frac{4\pi}{3}))\), let's go through the steps:

1. **Calculate \(\cos(\frac{4\pi}{3})\)**:
   \[
   \frac{4\pi}{3}\text{ radians} = 240^\circ
   \]
   The cosine of \(240^\circ\) (or \(\frac{4\pi}{3}\) radians) is \(-\frac{1}{2}\).

   So,
   \[
   \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}
   \]

2. **Evaluate \(\cos^{-1}(-\frac{1}{2})\)**:
   The function \(\cos^{-1}(x)\) (also called \(\arccos(x)\)) returns the angle in the range \( [0, \pi] \) whose cosine is \( x \).

   The angle \(\theta\) that satisfies \(\cos(\theta) = -\frac{1}{2}\) in the range \([0, \pi]\) is:
   \[
   \theta = \frac{2\pi}{3}
   \]

Therefore,
\[
\cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) = \frac{2\pi}{3}
\]

Would you like more details on this process or have any questions?

Here are five related questions you might consider:

1. What is the general form of the inverse trigonometric functions?
2. How do you determine the range of the inverse cosine function?
3. How would the evaluation differ if the angle was \(\frac{5\pi}{3}\)?
4. Can you derive the cosine of any angle using the unit circle?
5. How does the identity \(\cos^{-1}(\cos(x))\) change depending on the quadrant of \(x\)?

**Tip:** When dealing with inverse trigonometric functions, always remember the specific range for which the inverse is defined. For \(\cos^{-1}(x)\), this range is \([0, \pi]\).

Learn how to evaluate \(\cos^{-1}(\cos(\frac{4\pi}{3}))\) using trigonometric principles and the unit circle. Understand the range of inverse trigonometric functions and how to apply them to find angles within specified domains. Detailed step-by-step explanation provided.

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