The maximum values of `3 costheta+5sin(theta-(pi)/(6))` for any real value of `theta` is:

Options

A. `sqrt(1 9)`
B. `sqrt(79)/(2)`
C. `sqrt(31)`
D. `sqrt(34)`

Answer

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

### Step 1: Rewrite the expression using the sine subtraction formula
We know that:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
Using this, we can rewrite \(5 \sin \left( \theta - \frac{\pi}{6} \right)\) as:
\[ 5 \left( \sin \theta \cos \frac{\pi}{6} - \cos \theta \sin \frac{\pi}{6} \right) \]
Substituting the values of \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin \frac{\pi}{6} = \frac{1}{2}\), we get:
\[ 5 \sin \left( \theta - \frac{\pi}{6} \right) = 5 \left( \sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2} \right) \]
This simplifies to:
\[ \frac{5\sqrt{3}}{2} \sin \theta - \frac{5}{2} \cos \theta \]

### Step 2: Substitute back into the original expression
Now substituting this back into the original expression, we have:
\[ 3 \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta - \frac{5}{2} \cos \theta \]
Combining the terms involving \(\cos \theta\):
\[ \left(3 - \frac{5}{2}\right) \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta = \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta \]

### Step 3: Find the maximum value of the combined expression
The expression can be written as:
\[ \frac{1}{2} \cos \theta + \frac{5\sqrt{3}}{2} \sin \theta \]
The maximum value of \(a \cos \theta + b \sin \theta\) is given by \(\sqrt{a^2 + b^2}\). Here, \(a = \frac{1}{2}\) and \(b = \frac{5\sqrt{3}}{2}\).

Calculating \(a^2 + b^2\):
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{5\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{25 \cdot 3}{4} = \frac{1}{4} + \frac{75}{4} = \frac{76}{4} = 19 \]

### Step 4: Calculate the maximum value
Thus, the maximum value is:
\[ \sqrt{19} \]

### Conclusion
The maximum value of \(3 \cos \theta + 5 \sin \left( \theta - \frac{\pi}{6} \right)\) for any real value of \(\theta\) is:
\[ \sqrt{19} \]