If `a leq 3 cos (theta+pi/3)+5cos theta+3 leq b`, find `a and b`, where a is the minimum value & b is the If a maximum value.

Answer

To solve the problem, we need to find the minimum value \( a \) and the maximum value \( b \) of the expression \( 3 \cos(\theta + \frac{\pi}{3}) + 5 \cos \theta + 3 \).

### Step 1: Rewrite the expression
The expression can be rewritten using the cosine addition formula:
\[ \cos(\theta + \frac{\pi}{3}) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
Substituting the values \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we have:
\[ 3 \cos(\theta + \frac{\pi}{3}) = 3 \left( \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \right) = \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 2: Combine like terms
Now, we can combine this with the rest of the expression:
\[ \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 5 \cos \theta + 3 \]
Combining the cosine terms:
\[ \left( \frac{3}{2} + 5 \right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]

### Step 3: Identify coefficients
Let \( A = -\frac{3\sqrt{3}}{2} \) and \( B = \frac{13}{2} \). We can now express the function in the form:
\[ f(\theta) = A \sin \theta + B \cos \theta + 3 \]

### Step 4: Find the maximum and minimum values
The maximum and minimum values of \( A \sin \theta + B \cos \theta \) can be found using the formula:
\[ \sqrt{A^2 + B^2} \]
Calculating \( A^2 \) and \( B^2 \):
\[ A^2 = \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4}, \quad B^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
Now, adding these:
\[ A^2 + B^2 = \frac{27}{4} + \frac{169}{4} = \frac{196}{4} = 49 \]
Thus,
\[ \sqrt{A^2 + B^2} = \sqrt{49} = 7 \]

### Step 5: Determine maximum and minimum values
The maximum value of \( f(\theta) \) is:
\[ \text{Maximum} = 7 + 3 = 10 \]
The minimum value of \( f(\theta) \) is:
\[ \text{Minimum} = -7 + 3 = -4 \]

### Conclusion
Thus, the values of \( a \) and \( b \) are:
\[ a = -4, \quad b = 10 \]