Question

Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3θ cos θ
(ii) 2 cos 3θ sin 2θ
(iii) 2 sin 4θ sin 3θ
(iv) 2 cos 7θ cos 3θ

Answer 1

(i)

$$\begin{gathered} 2\left(\sin 3\theta \right)\left(\cos \theta \right) \\ =\sin \left(3\theta +\theta \right)+\sin \left(3\theta -\theta \right)\left[\because 2\sin AcosB=\sin (A+B)+\sin (A-B)\right] \\ =\sin 4\theta +\sin 2\theta \end{gathered}$$


(ii)

$$\begin{gathered} 2\left(\cos 3\theta \right)\left(\sin 2\theta \right) \\ =\sin \left(3\theta +2\theta \right)-\sin \left(3\theta -2\theta \right)\left[\because 2\cos A\sin B=\sin (A+B)-\sin (A-B)\right] \\ =\sin 5\theta -\sin \theta \end{gathered}$$


(iii)

$$\begin{gathered} 2\left(\sin 4\theta \right)\left(\sin 3\theta \right) \\ =\cos \left(4\theta -3\theta \right)-\cos \left(4\theta +3\theta \right)\left[\because 2\sin A\sin B=\cos (A-B)-\cos (A+B)\right] \\ =\cos \theta -\cos 7\theta \end{gathered}$$


(iv)

$$\begin{gathered} 2\left(\cos 7\theta \right)\left(\cos 3\theta \right) \\ =\cos \left(7\theta +3\theta \right)+\cos \left(7\theta -3\theta \right)\left[\because 2\cos A\cos B=\cos (A+B)+\cos (A-B)\right] \\ =\cos 10\theta +\cos 4\theta \end{gathered}$$