Question

The maximum value of $3cos\theta +5sin\left(\theta -\frac{\pi }{6}\right)$ for any real value of $\theta$ is :

• A. $\sqrt{19}$
• B. $\frac{\sqrt{79}}{2}$
• C. $\sqrt{31}$
• D. $\sqrt{34}$

Answer 1

Answer: (A)

$y=3\cos \theta +5\left(\sin \theta \frac{\sqrt{3}}{2}-\cos \theta \frac{1}{2}\right)$
$\frac{5\sqrt{3}}{2}\sin \theta +\frac{1}{2}\cos \theta$
$y_{\max }=\sqrt{\frac{75}{4}+\frac{1}{4}}=\sqrt{19}$