The maximum and minimum values of
`-4le5cos theta+3cos(theta+(pi)/(3))+3le10` are respectively

Options

A. 5
B. 10
C. 11
D. `-11`

Answer

To find the maximum and minimum values of the expression \(-4 \leq 5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right) + 3 \leq 10\), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the term \(3 \cos\left(\theta + \frac{\pi}{3}\right)\) using the cosine addition formula:
\[ \cos\left(a + b\right) = \cos a \cos b - \sin a \sin b \]
Thus, we have:
\[ 3 \cos\left(\theta + \frac{\pi}{3}\right) = 3 \left(\cos \theta \cos\left(\frac{\pi}{3}\right) - \sin \theta \sin\left(\frac{\pi}{3}\right)\right) \]
Using \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), we get:
\[ 3 \cos\left(\theta + \frac{\pi}{3}\right) = 3 \left(\cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2}\right) = \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 2: Combine the terms
Now we can combine the terms:
\[ 5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right) = 5 \cos \theta + \left(\frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta\right) \]
This simplifies to:
\[ \left(5 + \frac{3}{2}\right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 3: Find maximum and minimum values
To find the maximum and minimum values of the expression:
\[ \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]
we can use the formula for the maximum and minimum values of \(a \cos \theta + b \sin \theta\):
\[ \sqrt{a^2 + b^2} \]
where \(a = \frac{13}{2}\) and \(b = -\frac{3\sqrt{3}}{2}\).

Calculating \(a^2 + b^2\):
\[ \left(\frac{13}{2}\right)^2 + \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]
Thus, the maximum value is:
\[ \sqrt{49} = 7 \]
And the minimum value is:
\[ -\sqrt{49} = -7 \]

### Step 4: Include the constant term
Now we need to include the constant term \(+3\) from the original expression:
- Maximum value:
\[ 7 + 3 = 10 \]
- Minimum value:
\[ -7 + 3 = -4 \]

### Final Result
Thus, the maximum and minimum values of the expression are:
- Maximum value: \(10\)
- Minimum value: \(-4\)