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  4. The maximum value of 3cosθ + 5sin (θ - (π/6)) for any real value of θ is :

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

JEE MainJEE Main 2019Application of Derivatives

Solution:

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