Question

Expand each of the following:

(1) \(\cos (\mathrm{X}-\mathrm{Y})\)

(2) \(\sin \left(\mathrm{A}-20^{\circ}\right)\)

(3) \(\sin \left(2 \alpha+45^{\circ}\right)\)

Answer 1

(1) \(\cos (X-Y)=\cos X \cdot \cos Y+\sin X \cdot \sin Y\)

(2) \(\sin \left(\mathrm{A}-20^{\circ}\right)=\sin \mathrm{A} \cdot \cos 20^{\circ}-\cos \mathrm{A} \cdot \sin 20^{\circ}\)

(3) \(\sin \left(2 \alpha+45^{\circ}\right)=\sin 2 \alpha \cdot \cos 45^{\circ}+\cos 2 \alpha \cdot \sin 45^{\circ}\)
\[
\begin{aligned}
& =\sin 2 \alpha \cdot\left(\frac{\sqrt{2}}{2}\right)+\cos 2 \alpha \cdot\left(\frac{\sqrt{2}}{2}\right) \\
& =\left(\frac{\sqrt{2}}{2}\right)(\sin 2 \alpha+\cos 2 \alpha)=\frac{\sqrt{2}(\sin 2 \alpha+\cos 2 \alpha)}{2}
\end{aligned}
\]