To find the greatest and least value of the expression $f(x) = 7 + 3\sin(x) - 4\cos(x)$, we'll follow these steps:

Step 1: Express in a single trigonometric function

We can combine the terms involving $\sin(x)$ and $\cos(x)$ into a single sine term using a trigonometric identity.

We rewrite the expression $3\sin(x) - 4\cos(x)$ in the form: $R\sin(x + \alpha)$ where $R$ is the amplitude and $\alpha$ is a phase shift. This is done by finding $R$ and $\alpha$.

Step 2: Find $R$

The formula for $R$ is: $R = \sqrt{a^2 + b^2}$ where $a = 3$ and $b = -4$. So, $R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Step 3: Find $\alpha$

The angle $\alpha$ satisfies: $\cos(\alpha) = \frac{a}{R} = \frac{3}{5}, \quad \sin(\alpha) = \frac{b}{R} = \frac{-4}{5}$ So, $\alpha = \tan^{-1}\left(\frac{-4}{3}\right)$. This places $\alpha$ in the fourth quadrant.

Now, the expression becomes: $f(x) = 7 + 5\sin(x + \alpha)$

Step 4: Find the maximum and minimum values of $f(x)$

The sine function $\sin(x + \alpha)$ oscillates between $-1$ and $1$. Therefore, the expression $5\sin(x + \alpha)$ oscillates between $-5$ and $5$.

Thus, the expression $f(x) = 7 + 5\sin(x + \alpha)$ oscillates between: $7 - 5 = 2 \quad \text{(minimum value)}$ and $7 + 5 = 12 \quad \text{(maximum value)}$

Conclusion:

• The maximum value of $f(x)$ is $12$.
• The minimum value of $f(x)$ is $2$.

Would you like further details on any of the steps?

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Related Questions:

1. How do you derive the general solution for $\alpha$ when combining sine and cosine terms?
2. What are other ways to simplify trigonometric expressions involving sine and cosine?
3. How does the phase shift affect the graph of a sine or cosine function?
4. Can you explain how to graph $f(x) = 7 + 3\sin(x) - 4\cos(x)$ to find the range visually?
5. What is the general procedure for finding maximum and minimum values of trigonometric expressions?

Tip: To express sums of sine and cosine in a single sine function, always look for the amplitude and phase shift using $R = \sqrt{a^2 + b^2}$ and $\alpha = \tan^{-1}\left(\frac{b}{a}\right)$.