Question 17.2.11: Find the general solution for each of the following equation......
Find the general solution for each of the following equations:
(i) \cos 4 x=\cos 2 x \quad(ii) \cos 3 x=\sin 2 x
(iii) \sin 3 x+\cos 2 x=0 \quad(iv) \sin m x+\sin n x=0
(i) \cos 4 x=\cos 2 x \\\Rightarrow \cos 4 x-\cos 2 x=0\\
\begin{array}{l}\Rightarrow-2 \sin \frac{(4 x+2 x)}{2} \sin \frac{(4 x-2 x)}{2}=0 \\ \\\qquad\left[\because \cos C-\cos D=-2 \sin \frac{(C+D)}{2} \sin \frac{(C-D)}{2}\right] \\ \\\Rightarrow-2 \sin 3 x \sin x=0\\ \\\Rightarrow \sin 3 x=0 \text { or } \sin x=0 \\ \\\Rightarrow 3 x=n \pi \text { or } x=m \pi, \text { where } m, n \in I \\ \\\Rightarrow x=\frac{n \pi}{3} \text { or } x=m \pi, \text { where } m, n \in I .\end{array}Hence, the general solution is x=\frac{n \pi}{3} or x=m \pi , where m, n \in I .
(ii) \cos 3 x=\sin 2 x \\ \\ \Rightarrow \cos 3 x=\cos \left(\frac{\pi}{2}-2 x\right)\\ \\ \Rightarrow 3 x=2 n \pi \pm\left(\frac{\pi}{2}-2 x\right) \quad[\cos \theta=\cos \alpha \Rightarrow \theta=2 n \pi \pm \alpha] \\ \\ \Rightarrow 3 x=2 n \pi+\left(\frac{\pi}{2}-2 x\right) or 3 x=2 n \pi-\left(\frac{\pi}{2}-2 x\right) , where n \in I \\ \\ \Rightarrow 5 x=2 n \pi+\frac{\pi}{2} or x=\left(2 n \pi-\frac{\pi}{2}\right) , where n \in I \\ \\ \Rightarrow x=\left(\frac{2 n \pi}{5}+\frac{\pi}{10}\right) or x=\left(2 n \pi-\frac{\pi}{2}\right) , where n \in I .
Hence, the general solution is x=\left(\frac{2 n \pi}{5}+\frac{\pi}{10}\right) or x=\left(2 n \pi-\frac{\pi}{2}\right) , where n \in I .
(iii) \sin 3 x+\cos 2 x=0 \\ \\\Rightarrow \cos 2 x=-\sin 3 x=\cos \left(\frac{\pi}{2}+3 x\right)\\ \\
\Rightarrow \cos 2 x=\cos \left(\frac{\pi}{2}+3 x\right) \\ \\ \Rightarrow 2 x=2 n \pi \pm\left(\frac{\pi}{2}+3 x\right) , where n \in I \\ \\ \Rightarrow 2 x=2 n \pi+\left(\frac{\pi}{2}+3 x\right) or 2 x=2 n \pi-\left(\frac{\pi}{2}+3 x\right) , where n \in I \\ \\ \Rightarrow x=\left(-2 n \pi-\frac{\pi}{2}\right) or x=\left(\frac{2 n \pi}{5}-\frac{\pi}{10}\right) , where n \in I \\ \\ \Rightarrow x=\left(2 n \pi-\frac{\pi}{2}\right) or x=\left(\frac{2 n \pi}{5}-\frac{\pi}{10}\right) , where n \in I .
Hence, the general solution is x=\left(2 n \pi-\frac{\pi}{2}\right) or x=\left(\frac{2 n \pi}{5}-\frac{\pi}{10}\right) , where n \in I .
NOTE \left(-2 n \pi-\frac{\pi}{2}\right) and \left(2 n \pi-\frac{\pi}{2}\right) give the same result as n \in I .
(iv) \sin m x+\sin n x=0 \\ \\ \Rightarrow 2 \sin \frac{(m+n) x}{2} \cos \frac{(m-n) x}{2}=0
or \sin \frac{(m+n) x}{2}=0 \quad or \cos \frac{(m-n) x}{2}=0 \\ \\ \Rightarrow \frac{(m+n) x}{2}=p \pi or \frac{(m-n) x}{2}=(2 q+1) \frac{\pi}{2} , where p, q \in I \\ \\ \Rightarrow \quad x=\frac{2 p \pi}{(m+n)} or x=\frac{(2 q+1) \pi}{(m-n)} , where p, q \in I .
Hence, the general solution is x=\frac{2 p \pi}{(m+n)} or x=\frac{(2 q+1) \pi}{(m-n)} , where p, q \in I .