Find maximum and minimum values of `2sin(theta+(pi)/(6))+sqrt(3)cos(theta-(pi)/(6))`

Answer

To find the maximum and minimum values of the expression \(2\sin\left(\theta + \frac{\pi}{6}\right) + \sqrt{3}\cos\left(\theta - \frac{\pi}{6}\right)\), we can start by rewriting the expression using the sine and cosine addition formulas.

### Step-by-Step Solution

1. **Rewrite the Expression**:
We can use the sine and cosine addition formulas:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
Thus, we can rewrite:
\[ 2\sin\left(\theta + \frac{\pi}{6}\right) = 2\left(\sin\theta \cos\frac{\pi}{6} + \cos\theta \sin\frac{\pi}{6}\right) \]
\[ = 2\left(\sin\theta \cdot \frac{\sqrt{3}}{2} + \cos\theta \cdot \frac{1}{2}\right) \]
\[ = \sqrt{3}\sin\theta + \cos\theta \]

For the cosine term:
\[ \sqrt{3}\cos\left(\theta - \frac{\pi}{6}\right) = \sqrt{3}\left(\cos\theta \cos\frac{\pi}{6} + \sin\theta \sin\frac{\pi}{6}\right) \]
\[ = \sqrt{3}\left(\cos\theta \cdot \frac{\sqrt{3}}{2} + \sin\theta \cdot \frac{1}{2}\right) \]
\[ = \frac{3}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta \]

2. **Combine the Terms**:
Now, we can combine both parts:
\[ y = \sqrt{3}\sin\theta + \cos\theta + \frac{3}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta \]
\[ = \left(\sqrt{3} + \frac{\sqrt{3}}{2}\right)\sin\theta + \left(1 + \frac{3}{2}\right)\cos\theta \]
\[ = \frac{3\sqrt{3}}{2}\sin\theta + \frac{5}{2}\cos\theta \]

3. **Identify Coefficients**:
Let \(A = \frac{3\sqrt{3}}{2}\) and \(B = \frac{5}{2}\).

4. **Find Maximum and Minimum Values**:
The maximum and minimum values of the expression \(A\sin\theta + B\cos\theta\) can be found using the formula:
\[ R = \sqrt{A^2 + B^2} \]
where \(R\) is the amplitude of the sine-cosine function.

Calculate \(R\):
\[ R = \sqrt{\left(\frac{3\sqrt{3}}{2}\right)^2 + \left(\frac{5}{2}\right)^2} \]
\[ = \sqrt{\frac{27}{4} + \frac{25}{4}} = \sqrt{\frac{52}{4}} = \sqrt{13} \]

5. **Determine Maximum and Minimum Values**:
The maximum value is \(R = \sqrt{13}\) and the minimum value is \(-R = -\sqrt{13}\).

### Final Answer:
- Maximum value: \(\sqrt{13}\)
- Minimum value: \(-\sqrt{13}\)