Question

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

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

Answer 1

The correct option is B $\sqrt{19}$

Let $\mu =3\cos \theta +5\sin (\theta -\frac{\pi }{6})$
$=3\cos \theta +5\sin \theta .\cos \frac{\pi }{6}-5\cos \theta .\sin \frac{\pi }{6}$
$=\frac{1}{2}\cos \theta +\frac{5\sqrt{3}}{2}\sin \theta$
$\therefore \text{max}\mu =\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(\frac{5\sqrt{3}}{2}\right)}^2}$
$=\sqrt{19}$