For all values of `theta,3-costheta+cos(theta+(pi)/(3))` lie in the interval

Options

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

Answer

To solve the problem, we need to find the range of the expression \(3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right)\) for all values of \(\theta\).

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ 3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right) \]

2. **Use the cosine addition formula**:
The cosine addition formula states that:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
Applying this to \(\cos\left(\theta + \frac{\pi}{3}\right)\):
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos \theta \cos\left(\frac{\pi}{3}\right) - \sin \theta \sin\left(\frac{\pi}{3}\right) \]
Knowing that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), we substitute:
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \]

3. **Substitute back into the expression**:
\[ 3 - \cos \theta + \left(\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) \]
Combine like terms:
\[ 3 - \cos \theta + \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta = 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

4. **Rearranging the expression**:
\[ 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]
Let \(a = -\frac{1}{2}\) and \(b = -\frac{\sqrt{3}}{2}\).

5. **Find the range of the expression**:
The expression can be written in the form \(c + a \cos \theta + b \sin \theta\) where \(c = 3\). The maximum and minimum values of \(a \cos \theta + b \sin \theta\) can be found using the formula:
\[ R = \sqrt{a^2 + b^2} \]
Calculate \(R\):
\[ R = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

6. **Determine the range**:
The range of \(c + a \cos \theta + b \sin \theta\) is:
\[ c - R \leq 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \leq c + R \]
Substituting \(c\) and \(R\):
\[ 3 - 1 \leq 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \leq 3 + 1 \]
This simplifies to:
\[ 2 \leq 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \leq 4 \]

7. **Final result**:
Thus, the expression \(3 - \cos \theta + \cos\left(\theta + \frac{\pi}{3}\right)\) lies in the interval \([2, 4]\).