The maximum value of 3cos θ + 5sin (θ - (π)/(6)) for any real value...

Given $f\left(\theta\right) = 3cos \theta + 5\sin \left(\theta - \dfrac{\pi}{6}\right)$

$⇒ f\left(\theta\right) = 3cos \theta + 5\left(sin \theta \cdot \dfrac{\sqrt{3}}{2} - cos \theta \cdot \dfrac{1}{2}\right)$

$⇒ f\left(\theta\right) = \dfrac{5\sqrt{3}}{2}sin \theta + \dfrac{1}{2}cos \theta$

Now using the concept $acosx + bsinx + c \in \left[c - \sqrt{{a}^{2} + {b}^{2}}, c + \sqrt{{a}^{2} + {b}^{2}}\right]$, we can write

Maximum value of $f\left(\theta\right)$ is $\sqrt{{\left(\dfrac{5\sqrt{3}}{2}\right)}^{2} + {\left(\dfrac{1}{2}\right)}^{2}} = \sqrt{\dfrac{75}{4} + \dfrac{1}{4}} = \sqrt{19}$