Step 1: Understanding the Concept:
This question asks for the maximum value of a linear combination of sine and cosine functions of the form $a \cos\theta + b \sin\theta$.
Step 2: Key Formula or Approach:
For an expression of the form $f(\theta) = a \cos\theta + b \sin\theta$, the range of values is given by:
\[ -\sqrt{a^2 + b^2} \le a \cos\theta + b \sin\theta \le \sqrt{a^2 + b^2} \]
Therefore, the maximum value is $\sqrt{a^2 + b^2}$ and the minimum value is $-\sqrt{a^2 + b^2}$.
Step 3: Detailed Explanation:
In the given expression, $3 \cos\theta + 4 \sin\theta$, we have:
\[ a = 3 \]
\[ b = 4 \]
Using the formula for the maximum value:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} \]
\[ = \sqrt{3^2 + 4^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]
Step 4: Final Answer:
The maximum value of the expression $3 \cos\theta + 4 \sin\theta$ is 5. Therefore, option (C) is the correct answer.