Question

Express the following as a single trigonometric ratio:

(1) \(\sin 2 \theta \cdot \cos \theta+\cos 2 \theta \cdot \sin \theta\)

(2) \(\cos 70^{\circ} \cdot \cos x+\sin 70^{\circ} \cdot \sin x\)

(3) \(\cos x \cdot \sin 3 x-\cos 3 x \cdot \sin x\)

(4) \(\sin 3 \theta \cdot \sin 2 \theta-\cos 3 \theta \cdot \cos 2 \theta\)

Answer 1

(a) (1) \(\sin 2 \theta \cdot \cos \theta+\cos 2 \theta \cdot \sin \theta=\sin (2 \theta+\theta)=\sin 3 \theta\)
(2) \(\cos 70^{\circ} \cdot \cos x+\sin 70^{\circ} \cdot \sin x=\cos \left(70^{\circ}-x\right)\)
(3)
\[
\begin{array}{ll}
\cos x \cdot \sin 3 x-\cos 3 x \cdot \sin x & \\
=\sin 3 x \cdot \cos x-\cos 3 x \cdot \sin x & \text { (rewrite } \cos x \cdot \sin 3 x \text { as } \sin 3 x \cdot \cos x) \\
=\sin (3 x-x) & (\sin \mathrm{A} \cdot \cos \mathrm{B}-\cos \mathrm{A} \cdot \sin \mathrm{B}=\sin (\mathrm{A}-\mathrm{B})) \\
=\sin 2 x &
\end{array}
\]

(4)
\[
\begin{aligned}
& \sin 3 \theta \cdot \sin 2 \theta-\cos 3 \theta \cdot \cos 2 \theta \\
& =-(\cos 3 \theta \cdot \cos 2 \theta-\sin 3 \theta \cdot \sin 2 \theta) \\
& =-\cos (3 \theta+2 \theta)=-\cos 5 \theta \quad \quad(\cos A \cdot \cos B-\sin A \cdot \sin B=\cos (A+B))
\end{aligned}
\]