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 analyze the expression \(3 - \cos \theta + \cos \left( \theta + \frac{\pi}{3} \right)\) and determine the interval it lies in for all values of \(\theta\).

### Step-by-Step Solution:

**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 \(\cos \left( \theta + \frac{\pi}{3} \right)\):
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
Since \(\cos \frac{\pi}{3} = \frac{1}{2}\) and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\), we have:
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \]

**Step 2: Substitute back into the expression.**

Now substituting this back into our original expression:
\[ 3 - \cos \theta + \left( \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \right) \]
This simplifies to:
\[ 3 - \cos \theta + \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta = 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

**Step 3: Combine like terms.**

Now we can combine the cosine terms:
\[ 3 - \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

**Step 4: Identify the coefficients for the A cos θ + B sin θ form.**

Let \(A = -\frac{1}{2}\) and \(B = -\frac{\sqrt{3}}{2}\). The expression can be rewritten as:
\[ 3 + A \cos \theta + B \sin \theta \]

**Step 5: Find the range of \(A \cos \theta + B \sin \theta\).**

The maximum and minimum values of \(A \cos \theta + B \sin \theta\) can be found using the formula:
\[ \sqrt{A^2 + B^2} \]
Calculating \(A^2 + B^2\):
\[ A^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}, \quad B^2 = \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]
Thus,
\[ A^2 + B^2 = \frac{1}{4} + \frac{3}{4} = 1 \]
So,
\[ \sqrt{A^2 + B^2} = \sqrt{1} = 1 \]

**Step 6: Determine the range of the entire expression.**

The expression \(A \cos \theta + B \sin \theta\) will range from \(-1\) to \(1\). Therefore, the entire expression \(3 + A \cos \theta + B \sin \theta\) will range from:
\[ 3 - 1 \quad \text{to} \quad 3 + 1 \]
This gives us:
\[ 2 \quad \text{to} \quad 4 \]

### Conclusion:

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