Question

Reduce each of the following expressions to the sine and cosine of a single expression:
(i) $\sqrt{3}\sin x-\cos x$
(ii) cos x − sin x
(iii) 24 cos x + 7 sin x

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Let}f\left(x\right)=\sqrt{3}\sin x-\cos x \\ \mathrm{Dividing}\mathrm{and}\mathrm{multiplying}\mathrm{by}\sqrt{3+1},\mathrm{i}.\mathrm{e}.\mathrm{by}2,\mathrm{we}\mathrm{get}: \\ f\left(x\right)=2\left(\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x\right) \\ \Rightarrow f\left(x\right)=2\left(\cos \frac{\mathrm{\pi }}{6}\sin x-\sin \frac{\mathrm{\pi }}{6}\cos x\right) \\ \Rightarrow f\left(x\right)=2\sin \left(x-\frac{\mathrm{\pi }}{6}\right) \\ \mathrm{Again}, \\ f\left(x\right)=2\left(\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x\right) \\ \Rightarrow f\left(x\right)=2\left(\sin \frac{\mathrm{\pi }}{3}\sin x-\cos \frac{\mathrm{\pi }}{3}\cos x\right) \\ \Rightarrow f\left(x\right)=-2\cos \left(\frac{\mathrm{\pi }}{3}+x\right) \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}f\left(x\right)=\cos x-\sin x \\ \mathrm{Dividing}\mathrm{and}\mathrm{multiplying}\mathrm{by}\sqrt{1^2+1^2},\mathrm{i}.\mathrm{e}.\mathrm{by}\sqrt{2,}\mathrm{we}\mathrm{get}: \\ f\left(x\right)=\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x\right) \\ \Rightarrow f\left(x\right)=\sqrt{2}(\cos 45°\cos x-\sin 45°\sin x) \\ \Rightarrow f\left(x\right)=\sqrt{2}\cos \left(\frac{\mathrm{\pi }}{4}+x\right) \\ \mathrm{Again}, \\ f\left(x\right)=\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x\right) \\ \Rightarrow f\left(x\right)=\sqrt{2}(\sin 45°\cos x-\cos 45°\sin x) \\ \Rightarrow f\left(x\right)=\sqrt{2}\sin \left(\frac{\mathrm{\pi }}{4}-x\right) \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Let}f\left(x\right)=24\cos x+7\sin x \\ \mathrm{Dividing}\mathrm{and}\mathrm{multiplying}\mathrm{by}\sqrt{{24}^2+7^2},\mathrm{i}.\mathrm{e}.\mathrm{by}25,\mathrm{we}\mathrm{get}: \\ f\left(x\right)=25\left(\frac{24}{25}\cos x+\frac{7}{25}\sin x\right) \\ \Rightarrow f\left(x\right)=25(\sin \alpha \cos x+\cos \alpha \sin x),\mathrm{where}\sin \alpha =\frac{24}{25}\mathrm{and}\cos \alpha =\frac{7}{25} \\ \Rightarrow f\left(x\right)=25\sin (\alpha +x),\mathrm{where}\tan \alpha =\frac{24}{7}. \\ \mathrm{Again}, \\ f\left(x\right)=25\left(\frac{24}{25}\cos x+\frac{7}{25}\sin x\right) \\ \Rightarrow f\left(x\right)=25(\cos \alpha \cos x+\sin \alpha \sin x),\mathrm{where}\cos \alpha =\frac{24}{25},\sin \alpha =\frac{7}{25}. \\ \Rightarrow f\left(x\right)=25\cos (\alpha -x),\mathrm{where}\tan \alpha =\frac{7}{24}. \end{gathered}$$