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

Options

A. `and -4`
B. `10 and -4`
C. `10 and -10`
D. `6and -4`

Explanation

To solve the problem of finding 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 can break it down step by step.

### Step 1: Rewrite the expression
We need to express \(3 \cos\left(\theta + \frac{\pi}{3}\right)\) using the cosine addition formula:

\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]

Substituting the values of \(\cos \frac{\pi}{3} = \frac{1}{2}\) and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\):

\[ 3 \cos\left(\theta + \frac{\pi}{3}\right) = 3\left(\cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2}\right) = \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 2: Combine the terms
Now, substituting this back into the original expression:

\[ 5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right) = 5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

Combining the cosine terms:

\[ = \left(5 + \frac{3}{2}\right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 3: Identify the maximum and minimum values
The expression can be treated as \(A \cos \theta + B \sin \theta\) where:

- \(A = \frac{13}{2}\)
- \(B = -\frac{3\sqrt{3}}{2}\)

The maximum and minimum values of \(A \cos \theta + B \sin \theta\) can be found using the formula:

\[ \sqrt{A^2 + B^2} \]

Calculating \(A^2 + B^2\):

\[ A^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
\[ B^2 = \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]

Adding these:

\[ A^2 + B^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]

Thus:

\[ \sqrt{A^2 + B^2} = \sqrt{49} = 7 \]

### Step 4: Determine the maximum and minimum values
The maximum value of \(5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right)\) is \(7\) and the minimum value is \(-7\).

### Step 5: Adjust for the inequality
Now, considering the original inequality:

\[ -4 \leq 5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right) + 3 \leq 10 \]

Adding \(3\) to the maximum and minimum values:

- Maximum: \(7 + 3 = 10\)
- Minimum: \(-7 + 3 = -4\)

Thus, the maximum and minimum values of the expression \(5 \cos \theta + 3 \cos\left(\theta + \frac{\pi}{3}\right) + 3\) are:

- Maximum value: \(10\)
- Minimum value: \(-4\)

### Final Answer:
The maximum and minimum values are respectively \(10\) and \(-4\).

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