To find the maximum value of the expression $3 + \sin x$:

Step-by-step solution:

1. Understand the range of $\sin x$: The sine function ($\sin x$) has a range of values between $-1$ and $1$. That is: $-1 \leq \sin x \leq 1$

2. Add 3 to the range: Adding 3 to each value of $\sin x$ shifts the range up by 3 units: $3 - 1 \leq 3 + \sin x \leq 3 + 1$ $2 \leq 3 + \sin x \leq 4$

3. Maximum value: The maximum value of $3 + \sin x$ is: $\boxed{4}$

4. When does this occur? The maximum value of $3 + \sin x$ occurs when $\sin x = 1$. This happens at: $x = \frac{\pi}{2} + 2n\pi, \quad n \in \mathbb{Z} \quad (\text{any integer multiple of the sine period}).$

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Related questions:

1. What is the minimum value of $3 + \sin x$?
2. What is the range of $3 + 2\sin x$?
3. How does the period of $\sin x$ affect the maximum and minimum points of $3 + \sin x$?
4. Can we generalize the result for $a + b\sin x$? What would its maximum be?
5. What are the key trigonometric transformations that affect the sine function's range?

Tip:

Always think about the amplitude and vertical shift of the trigonometric function when analyzing expressions like $a + b\sin x$.