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.