Find the greatest and least value of 7+3sinx-4cosx

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 \( \alpha = \tan^{-1}\left(\frac{b}{a}\right) \).

Find the greatest and least value of 7 + 3sin(x) - 4cos(x)

Discover how to find the greatest and least values of the expression 7 + 3sin(x) - 4cos(x). This step-by-step solution simplifies trigonometric expressions using amplitude and phase shift, suitable for Grades 10-12.

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