Find the maximum and minimum value of `5cos theta+3sin(theta+(pi)/(6))` for all real values of `theta`. .

Answer

To find the maximum and minimum values of the expression \( 5 \cos \theta + 3 \sin\left(\theta + \frac{\pi}{6}\right) \), we can follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the sine term using the sine addition formula:
\[ \sin\left(\theta + \frac{\pi}{6}\right) = \sin\theta \cos\frac{\pi}{6} + \cos\theta \sin\frac{\pi}{6} \]
Substituting the values of \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin\frac{\pi}{6} = \frac{1}{2}\), we have:
\[ \sin\left(\theta + \frac{\pi}{6}\right) = \sin\theta \cdot \frac{\sqrt{3}}{2} + \cos\theta \cdot \frac{1}{2} \]

### Step 2: Substitute back into the expression
Now substitute this back into the original expression:
\[ 5 \cos \theta + 3 \left( \sin\theta \cdot \frac{\sqrt{3}}{2} + \cos\theta \cdot \frac{1}{2} \right) \]
This simplifies to:
\[ 5 \cos \theta + \frac{3\sqrt{3}}{2} \sin \theta + \frac{3}{2} \cos \theta \]
Combining the cosine terms:
\[ \left(5 + \frac{3}{2}\right) \cos \theta + \frac{3\sqrt{3}}{2} \sin \theta = \frac{13}{2} \cos \theta + \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 3: Identify coefficients
Let \( a = \frac{3\sqrt{3}}{2} \) and \( b = \frac{13}{2} \). We can express the equation as:
\[ a \sin \theta + b \cos \theta \]

### Step 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:
- Maximum value: \( \sqrt{a^2 + b^2} \)
- Minimum value: \( -\sqrt{a^2 + b^2} \)

Calculating \( a^2 + b^2 \):
\[ a^2 = \left(\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]
\[ b^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
Thus,
\[ a^2 + b^2 = \frac{27}{4} + \frac{169}{4} = \frac{196}{4} = 49 \]

### Step 5: Calculate the square root
Now, taking the square root:
\[ \sqrt{49} = 7 \]

### Step 6: State the maximum and minimum values
Therefore, the maximum value is \( 7 \) and the minimum value is \( -7 \).

### Final Answer:
- Maximum value: \( 7 \)
- Minimum value: \( -7 \)