Question

Simplify $5cos\theta +3cos\left(\theta +\frac{\pi }{3}\right)+3$

Answer 1

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\cos \frac{\pi }{3}=\frac{1}{2},\sin \pi /3=\frac{\sqrt{3}}{2}$

$\Rightarrow 5\cos \theta +3[\cos \frac{\pi }{3}\cos \theta -\sin \frac{\pi }{3}\sin \theta ]+3$

$\Rightarrow 5\cos \theta +3[\frac{\cos \theta }{2}-\frac{\sqrt{3}\sin \theta }{2}]+3$

$\Rightarrow 5\cos \theta +\frac{3\cos \theta }{2}-\frac{3\sqrt{3}}{2}\sin \theta +3$

$\Rightarrow \frac{13\cos \theta }{2}-\frac{3\sqrt{3}}{2}\sin \theta +3$