find the maximum and the minimum value of the expression 2-(2/3)*sinx

To find the maximum and minimum values of the expression \( f(x) = 2 - \frac{2}{3} \sin(x) \), we can proceed as follows:

### Step 1: Analyze the Sine Function
The sine function, \( \sin(x) \), varies between \(-1\) and \(1\):

\[
-1 \leq \sin(x) \leq 1
\]

### Step 2: Analyze the Term Involving \( \sin(x) \)
The expression \( f(x) \) involves the term \( \frac{2}{3} \sin(x) \):

\[
-\frac{2}{3} \leq \frac{2}{3} \sin(x) \leq \frac{2}{3}
\]

### Step 3: Add 2 to All Sides
The function \( f(x) = 2 - \frac{2}{3} \sin(x) \) is obtained by subtracting \( \frac{2}{3} \sin(x) \) from 2:

\[
2 - \frac{2}{3} \leq f(x) \leq 2 + \frac{2}{3}
\]

Simplify the above expressions:

\[
\frac{4}{3} \leq f(x) \leq \frac{8}{3}
\]

### Conclusion:
- The **minimum value** of the function is \( \frac{4}{3} \approx 1.333 \).
- The **maximum value** of the function is \( \frac{8}{3} \approx 2.667 \).

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

### Relative Questions:
1. How does adding or subtracting constants affect the range of a trigonometric function?
2. Can you explain why the function \( f(x) \) achieves its maximum and minimum values at specific points?
3. What happens to the function \( f(x) = 2 - \frac{2}{3} \sin(x) \) if we replace \( \sin(x) \) with \( \cos(x) \)?
4. How would the maximum and minimum values change if the coefficient of \( \sin(x) \) were different?
5. Can you determine the exact values of \( x \) where the maximum and minimum occur?

### Tip:
When working with trigonometric functions, understanding how the coefficients and constants modify the function's range is crucial for quickly determining its extrema.

Discover how to find the maximum and minimum values of the expression 2 - (2/3)*sin(x). This problem involves analyzing the sine function's range and understanding how it affects the overall function's extrema. Suitable for advanced high school students and anyone interested in trigonometric function analysis.

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