The range of vlaues of the expression `5 cos theta + 3 cos (theta + (pi)/(3)) +1` is

Options

A. `[-7,7]`
B. `[-6,8]`
C. `[-8,6]`
D. `[-3,sqrt3,13]`

Answer

To find the range of the expression \(5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 1\), we will follow these steps:

### Step 1: Rewrite the expression using the cosine addition formula
The cosine addition formula states that:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
Applying this to \(3 \cos \left(\theta + \frac{\pi}{3}\right)\):
\[ \cos \left(\theta + \frac{\pi}{3}\right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
We know that \(\cos \frac{\pi}{3} = \frac{1}{2}\) and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Thus:
\[ 3 \cos \left(\theta + \frac{\pi}{3}\right) = 3 \left(\cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2}\right) = \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 2: Substitute back into the original expression
Now substituting this back into the expression:
\[ 5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 1 = 5 \cos \theta + \left(\frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta\right) + 1 \]
Combine the cosine terms:
\[ = \left(5 + \frac{3}{2}\right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 1 \]
\[ = \frac{10}{2} \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 1 \]
\[ = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 1 \]

### Step 3: Simplify the expression
Now we can express this as:
\[ = \frac{1}{2}(13 \cos \theta - 3\sqrt{3} \sin \theta + 2) \]

### Step 4: Find the maximum and minimum values
To find the maximum and minimum values of \(13 \cos \theta - 3\sqrt{3} \sin \theta\), we can use the formula:
\[ R = \sqrt{a^2 + b^2} \]
where \(a = 13\) and \(b = -3\sqrt{3}\).

Calculating \(R\):
\[ R = \sqrt{13^2 + (-3\sqrt{3})^2} = \sqrt{169 + 27} = \sqrt{196} = 14 \]

The maximum value of \(13 \cos \theta - 3\sqrt{3} \sin \theta\) is \(R = 14\) and the minimum value is \(-R = -14\).

### Step 5: Adjust for the constant term
Now, substituting back into our expression:
- Maximum value:
\[ \frac{1}{2}(14 + 2) = \frac{16}{2} = 8 \]
- Minimum value:
\[ \frac{1}{2}(-14 + 2) = \frac{-12}{2} = -6 \]

### Conclusion
Thus, the range of the expression \(5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 1\) is:
\[ [-6, 8] \]