The maximum value of `5costheta+3cos(theta+pi/3)+3` is:

Options

A. 5
B. 11
C. 10
D. None of these

Answer

To find the maximum value of the expression \(5 \cos \theta + 3 \cos(\theta + \frac{\pi}{3}) + 3\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ f(\theta) = 5 \cos \theta + 3 \cos(\theta + \frac{\pi}{3}) + 3 \]

### Step 2: Expand the cosine term
Using the cosine addition formula, we can expand \( \cos(\theta + \frac{\pi}{3}) \):
\[ \cos(\theta + \frac{\pi}{3}) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
Substituting the values \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \):
\[ \cos(\theta + \frac{\pi}{3}) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \]

### Step 3: Substitute back into the expression
Now substituting this back into our function:
\[ f(\theta) = 5 \cos \theta + 3 \left(\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) + 3 \]
This simplifies to:
\[ f(\theta) = 5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
Combining the cosine terms:
\[ f(\theta) = \left(5 + \frac{3}{2}\right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
\[ f(\theta) = \frac{10 + 3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
\[ f(\theta) = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]

### Step 4: Identify coefficients for maximum value calculation
Let \( a = \frac{13}{2} \) and \( b = -\frac{3\sqrt{3}}{2} \). The maximum value of the function \( f(\theta) = a \cos \theta + b \sin \theta + 3 \) can be found using the formula:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} + 3 \]

### Step 5: Calculate \( a^2 + b^2 \)
Calculating \( a^2 \) and \( b^2 \):
\[ a^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
\[ b^2 = \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]
Now, adding these:
\[ a^2 + b^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]

### Step 6: Find the maximum value
Taking the square root:
\[ \sqrt{49} = 7 \]
Thus, the maximum value of \( f(\theta) \) is:
\[ 7 + 3 = 10 \]

### Final Answer
The maximum value of \( 5 \cos \theta + 3 \cos(\theta + \frac{\pi}{3}) + 3 \) is:
\[ \boxed{10} \]