Question

The maximum value of $3\cos \theta +5\sin (\theta -\frac{\pi }{6})$ 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)

$\sqrt{19}$

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