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The question provided is asking to find the value of \(\cos^{-1}(\cos 5r)\), where \(r\) is some variable.

To solve this, we use the property of the inverse cosine function \(\cos^{-1}(\cos \theta)\). The general solution for \(\cos^{-1}(\cos \theta)\) is:

\[
\cos^{-1}(\cos \theta) = \theta \quad \text{if} \quad 0 \leq \theta \leq \pi
\]

For angles beyond this range, we need to adjust \(\theta\) to fall within the principal range of the inverse cosine function, which is \([0, \pi]\).

Given \(\theta = 5r\), we know that this value can be outside the principal range, so we use the following rule:

\[
\cos^{-1}(\cos \theta) = 2\pi - \theta \quad \text{if} \quad \theta > \pi
\]

For this problem, since \(5r\) is not explicitly defined, let's assume \(5r\) is greater than \(\pi\). We can adjust the angle accordingly to fit within the principal range of the inverse cosine function. Therefore, the answer can be in the form of \(2\pi - 5r\).

Thus, the correct answer to the problem is:

**(D) \(2\pi - 5r\)**

Let me know if you have any questions or need further clarification!

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Here are 5 relative questions to further explore this concept:
1. What is the general solution for \(\cos^{-1}(\cos \theta)\) when \(\theta\) is outside the range \([0, \pi]\)?
2. How does the inverse trigonometric function differ from the regular trigonometric function in terms of output values?
3. What is the significance of the principal range in inverse trigonometric functions?
4. How can we adjust angles for other inverse trigonometric functions like \(\sin^{-1}\) or \(\tan^{-1}\)?
5. What are some practical applications of inverse trigonometric functions in real-world problems?

**Tip:** When working with inverse trigonometric functions, always remember to check if the angle falls within the principal range, and adjust accordingly to get the correct value!

Understand how to solve the inverse cosine problem cos^{-1}(cos 5r). This detailed solution explains the key concepts of inverse trigonometric functions and how to adjust angles to fit within the principal range. Suitable for students in Grades 11-12.

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