Find the maximum and minimum value of `cos^(2) theta- 6 sin theta cos theta + 3 sin^(2) theta + 2`.

Answer

To find the maximum and minimum values of the expression \( f(\theta) = \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \), we can follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We can use the identities for \(\cos^2 \theta\) and \(\sin^2 \theta\):
\[ \cos^2 \theta = \frac{1 + \cos 2\theta}{2}, \quad \sin^2 \theta = \frac{1 - \cos 2\theta}{2}, \quad \text{and} \quad \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \]

Substituting these into the expression:
\[ f(\theta) = \frac{1 + \cos 2\theta}{2} - 6 \left(\frac{1}{2} \sin 2\theta\right) + 3 \left(\frac{1 - \cos 2\theta}{2}\right) + 2 \]

### Step 2: Simplify the expression
Now, simplify the expression:
\[ f(\theta) = \frac{1 + \cos 2\theta}{2} - 3 \sin 2\theta + \frac{3 - 3 \cos 2\theta}{2} + 2 \]
Combining like terms:
\[ f(\theta) = \frac{1 + 3 + 2}{2} + \frac{\cos 2\theta - 3 \cos 2\theta}{2} - 3 \sin 2\theta \]
\[ = \frac{6}{2} - \frac{2 \cos 2\theta}{2} - 3 \sin 2\theta \]
\[ = 3 - \cos 2\theta - 3 \sin 2\theta \]

### Step 3: Rearranging the expression
Rearranging gives:
\[ f(\theta) = 3 - (\cos 2\theta + 3 \sin 2\theta) \]

### Step 4: Finding the maximum and minimum values
Let \( S = \cos 2\theta + 3 \sin 2\theta \). The maximum and minimum values of \( S \) can be found using the formula:
\[ \text{Maximum value} = \sqrt{A^2 + B^2}, \quad \text{Minimum value} = -\sqrt{A^2 + B^2} \]
where \( A = 1 \) and \( B = 3 \):
\[ \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

Thus, the maximum value of \( S \) is \( \sqrt{10} \) and the minimum value is \( -\sqrt{10} \).

### Step 5: Finding the maximum and minimum of \( f(\theta) \)
Now, substituting back into \( f(\theta) \):
- Maximum value of \( f(\theta) \):
\[ f_{\text{max}} = 3 - (-\sqrt{10}) = 3 + \sqrt{10} \]
- Minimum value of \( f(\theta) \):
\[ f_{\text{min}} = 3 - \sqrt{10} \]

### Final Answer
The maximum value is \( 3 + \sqrt{10} \) and the minimum value is \( 3 - \sqrt{10} \).

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