If `theta` is real then `3 - cos theta + cos (theta + (pi)/(3))` lies in the interval

Options

A. `[-2, 3]`
B. `[-2, 1]`
C. `[2, 4]`
D. `[1, 5]`

Answer

To solve the problem, we need to analyze the expression \(3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right)\) and determine the interval in which it lies.

### Step 1: Expand the expression
We start with the expression:
\[ 3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right) \]
Using the cosine addition formula, we can expand \(\cos\left(\theta + \frac{\pi}{3}\right)\):
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos \theta \cos\frac{\pi}{3} - \sin \theta \sin\frac{\pi}{3} \]
Substituting \(\cos\frac{\pi}{3} = \frac{1}{2}\) and \(\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\), we get:
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \]

### Step 2: Substitute back into the expression
Now substituting this back into our original expression:
\[ 3 - \cos \theta + \left(\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) \]
Combining like terms:
\[ 3 - \cos \theta + \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta = 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

### Step 3: Rewrite the expression
Now we can rewrite this as:
\[ 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

### Step 4: Identify coefficients for the interval calculation
Let’s denote:
- \(a = -\frac{1}{2}\)
- \(b = -\frac{\sqrt{3}}{2}\)
- \(c = 3\)

### Step 5: Calculate the range
We can find the maximum and minimum values of the expression \(c - \sqrt{a^2 + b^2}\) and \(c + \sqrt{a^2 + b^2}\).

Calculating \(a^2 + b^2\):
\[ a^2 + b^2 = \left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1 \]
Thus, \(\sqrt{a^2 + b^2} = \sqrt{1} = 1\).

### Step 6: Find the minimum and maximum values
Now we can find the minimum and maximum values:
- Minimum value:
\[ c - \sqrt{a^2 + b^2} = 3 - 1 = 2 \]
- Maximum value:
\[ c + \sqrt{a^2 + b^2} = 3 + 1 = 4 \]

### Conclusion
The expression \(3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right)\) lies in the interval \([2, 4]\).