Question

Solve the following equations:
(i) $\cos \mathrm{\theta }+\cos 2\mathrm{\theta }+\cos 3\mathrm{\theta }=0$
(ii) $\cos \mathrm{\theta }+\cos 3\mathrm{\theta }-\cos 2\mathrm{\theta }=0$
(iii) $\sin \mathrm{\theta }+\sin 5\mathrm{\theta }=\sin 3\mathrm{\theta }$
(iv) $\cos \mathrm{\theta }\cos 2\mathrm{\theta }\cos 3\mathrm{\theta }=\frac{1}{4}$
(v) $\cos \mathrm{\theta }+\sin \mathrm{\theta }=\cos 2\mathrm{\theta }+\sin 2\mathrm{\theta }$
(vi) $\sin \mathrm{\theta }+\sin 2\mathrm{\theta }+\sin 3=0$
(vii) $\sin \mathrm{\theta }+\sin 2\mathrm{\theta }+\sin 3\mathrm{\theta }+\sin 4\mathrm{\theta }=0$
(viii) $\sin 3\mathrm{\theta }-\sin \mathrm{\theta }=4{\cos }^2\mathrm{\theta }-2$
(ix) $\sin 2\mathrm{\theta }-\sin 4\mathrm{\theta }+\sin 6\mathrm{\theta }=0$

Answer 1

(i) $\cos \theta +\cos 2\theta +\cos 3\theta =0$
Now,
$$\begin{gathered} (\cos \theta +\cos 3\theta )+\cos 2\theta =0 \\ \Rightarrow 2\cos \left(\frac{4\theta }{2}\right)\cos \left(\frac{2\theta }{2}\right)+\cos 2\theta =0 \\ \Rightarrow 2\cos 2\theta \cos \theta +\cos 2\theta =0 \\ \Rightarrow \cos 2\theta (2\cos \theta +1)=0 \end{gathered}$$

$\Rightarrow \cos 2\theta =0$ or, $2\cos \theta +1=0$
$\Rightarrow \cos 2\theta =\cos \frac{\mathrm{\pi }}{2}$ or $\cos \theta =-\frac{1}{2}=\cos \frac{2\mathrm{\pi }}{3}$
$\Rightarrow 2\theta =(2n+1)\frac{\mathrm{\pi }}{2}$, $n\in Z$ or $\theta =2m\pi \pm \frac{2\mathrm{\pi }}{3},m\in Z$
$\Rightarrow \theta =(2n+1)\frac{\mathrm{\pi }}{4},n\in Z$ or $\theta =2m\pi \pm \frac{2\mathrm{\pi }}{3},m\in Z$

(ii)

$$\begin{gathered} (\cos \theta +\cos 3\theta )-\cos 2\theta =0 \\ \Rightarrow 2\cos \left(\frac{4\theta }{2}\right)\cos \left(\frac{2\theta }{2}\right)-\cos 2\theta =0 \\ \Rightarrow 2\cos 2\theta \cos \theta -\cos 2\theta =0 \\ \Rightarrow \cos 2\theta (2\cos \theta -1)=0 \end{gathered}$$

$\Rightarrow \cos 2\theta =0$ or $2\cos \theta -1=0$
$\Rightarrow \cos 2\theta =\cos \frac{\mathrm{\pi }}{2}$ or $\cos \theta =\frac{1}{2}\Rightarrow \cos \theta =\cos \frac{\mathrm{\pi }}{3}$

$\Rightarrow 2\theta =(2n+1)\frac{\mathrm{\pi }}{2},n\in Z$ or $\theta =2m\pi \pm \frac{\mathrm{\pi }}{3},m\in Z$

$\Rightarrow \theta =(2n+1)\frac{\mathrm{\pi }}{4},n\in Z$ or $\theta =2m\pi \pm \frac{\mathrm{\pi }}{3},m\in Z$

(iii) $\sin \theta +\sin 5\theta =\sin 3\theta$
$$\begin{gathered} \Rightarrow 2\sin \left(\frac{6\theta }{2}\right)\cos \left(\frac{4\theta }{2}\right)=\sin 3\theta \\ \Rightarrow 2\sin 3\theta \cos 2\theta =\sin 3\theta \\ \Rightarrow 2\sin 3\theta \cos 2\theta -\sin 3\theta =0 \\ \Rightarrow \sin 3\theta (2\cos 2\theta -1)=0 \end{gathered}$$

$\Rightarrow \sin 3\theta =0$ or $(2\cos 2\theta -1)=0$
$\Rightarrow \sin 3\theta =\sin 0$ or $\cos 2\theta =\frac{1}{2}=\cos \frac{\mathrm{\pi }}{3}$
$\Rightarrow 3\theta =n\pi$ or $2\theta =2m\pi \pm \frac{\mathrm{\pi }}{3}$
$\Rightarrow$ $\theta =\frac{n\mathrm{\pi }}{3},n\in Z$ or $\theta =m\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6},m\in \mathit{}Z$

$$\begin{gathered} \left(\mathrm{iv}\right)\cos \theta \cos 2\theta \cos 3\theta =\frac{1}{4} \\ \Rightarrow \left[\frac{\cos \left(\theta +2\theta \right)+\cos \left(2\theta -\theta \right)}{2}\right]\cos 3\theta =\frac{1}{4} \\ \Rightarrow 2\left[\cos 3\theta +\cos \theta \right]\cos 3\theta =1 \\ \Rightarrow 2{\cos }^{2}3\theta +2\cos \theta \cos 3\theta -1=0 \\ \Rightarrow 2{\cos }^{2}3\theta -1+2\cos \theta \cos 3\theta =0 \\ \Rightarrow \cos 6\theta +\cos 4\theta +\cos 2\theta =0 \\ \Rightarrow \cos 6\theta +\cos 2\theta +\cos 4\theta =0 \\ \Rightarrow 2\cos 4\theta \cos 2\theta +\cos 4\theta =0 \\ \Rightarrow \cos 4\theta \left(2\cos 2\theta +1\right)=0 \\ \Rightarrow \cos 4\theta =0\mathrm{or}2\cos 2\theta +1=0 \\ \Rightarrow \cos 4\theta =0\mathrm{or}\cos 2\theta =\frac{-1}{2} \\ \Rightarrow \cos 4\theta =\cos \frac{\mathrm{\pi }}{2}\mathrm{or}\cos 2\mathrm{\theta }=\cos \frac{2\mathrm{\pi }}{3} \\ \Rightarrow 4\theta =\left(2n+1\right)\frac{\mathrm{\pi }}{2},n\in Z\mathrm{or}2\theta =2m\mathrm{\pi }\pm \frac{2\mathrm{\pi }}{3},m\in Z \\ \Rightarrow \theta =\left(2n+1\right)\frac{\mathrm{\pi }}{8},n\in Z\mathrm{or}\theta =m\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3},m\in Z \end{gathered}$$


(v) $\cos \theta +\sin \theta =\cos 2\theta +\sin 2\theta$

$$\begin{gathered} \Rightarrow \cos \theta -\cos 2\theta =\sin 2\theta -\sin \theta \\ \Rightarrow -2\sin \left(\frac{3\theta }{2}\right)\sin \left(\frac{-\theta }{2}\right)=2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{3\theta }{2}\right) \\ \Rightarrow 2\sin \left(\frac{3\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)=2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{3\theta }{2}\right) \\ \Rightarrow 2\sin \left(\frac{\theta }{2}\right)\left[\sin \left(\frac{3\theta }{2}\right)-\cos \left(\frac{3\theta }{2}\right)\right]=0 \end{gathered}$$

$\Rightarrow \sin \frac{\theta }{2}=0$ or $\sin \frac{3\theta }{2}-\cos \frac{3\theta }{2}=0$

$\Rightarrow \sin \frac{\theta }{2}=\sin 0$ or $\sin \frac{3\theta }{2}=\cos \frac{3\theta }{2}$

$\Rightarrow$ $\frac{\theta }{2}=n\pi$, $n\in Z$ or $\cos \frac{3\theta }{2}=\cos \left(\frac{\mathrm{\pi }}{2}-\frac{3\theta }{2}\right)$

$\Rightarrow \theta =2n\pi ,n\in Z$ or $\frac{3\theta }{2}=2m\pi \pm \left(\frac{\mathrm{\pi }}{2}-\frac{3\theta }{2}\right),m\in Z$

$\Rightarrow$$\theta =2n\pi ,n\in Z$ or $\frac{3\theta }{2}=2m\pi +\frac{\mathrm{\pi }}{2}-\frac{3\theta }{2},m\in Z$ (Taking negative sign will give absurd result.)

$\theta =2n\pi ,n\in Z$ or $\theta =\frac{2m\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{6},m\in Z$


(vi) $\sin \theta +\sin 2\theta +\sin 3\theta =0$

$$\begin{gathered} \Rightarrow \sin \theta +\sin 3\theta +\sin 2\theta =0 \\ \Rightarrow 2\sin \left(\frac{4\theta }{2}\right)\cos \left(\frac{2\theta }{2}\right)+\sin 2\theta =0 \\ \Rightarrow 2\sin 2\theta \cos \theta +\sin 2\theta =0 \\ \Rightarrow \sin 2\theta (2\cos \theta +1)=0 \end{gathered}$$

$\Rightarrow \sin 2\theta =0$ or $2\cos \theta +1=0$
$\Rightarrow \sin 2\theta =\sin 0$ or $\cos \theta =-\frac{1}{2}\Rightarrow \cos \theta =\cos \frac{2\mathrm{\pi }}{3}$
$\Rightarrow$$\theta =\frac{n\mathrm{\pi }}{2},n\in Z$ or $\theta =2m\pi \pm \frac{2\mathrm{\pi }}{3}$, $m\in Z$

(vii) $\sin \theta +\sin 2\theta +\sin 3\theta +\sin 4\theta =0$

$$\begin{gathered} \Rightarrow \sin 3\theta +\sin \theta +\sin 4\theta +\sin 2\theta =0 \\ \Rightarrow 2\sin \left(\frac{4\theta }{2}\right)\cos \left(\frac{2\theta }{2}\right)+2\sin \left(\frac{6\theta }{2}\right)\cos \left(\frac{2\theta }{2}\right)=0 \\ \Rightarrow 2\sin 2\theta \cos \theta +2\sin 3\theta \cos \theta =0 \\ \Rightarrow 2\cos \theta (\sin 2\theta +\sin 3\theta )=0 \\ \Rightarrow 2\cos \theta \left(2\sin \left(\frac{5\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)\right)=0 \\ \Rightarrow 4\cos \theta \sin \left(\frac{5\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)=0 \end{gathered}$$
$\Rightarrow \cos \theta =0,\sin \left(\frac{5\theta }{2}\right)=0$ or $\cos \left(\frac{\theta }{2}\right)=0$
$\Rightarrow \cos \theta =\cos \frac{\mathrm{\pi }}{2},\sin \left(\frac{5\theta }{2}\right)=\sin 0$ or $\cos \left(\frac{\theta }{2}\right)=\cos \frac{\mathrm{\pi }}{2}$
$\Rightarrow \theta =(2n+1)\frac{\mathrm{\pi }}{2},n\in Zor\frac{5\theta }{2}=n\pi ,n\in Z$ or, $\frac{\theta }{2}=(2n+1)\frac{\mathrm{\pi }}{2},n\in Z$
$\Rightarrow \theta =(2n+1)\frac{\mathrm{\pi }}{2},n\in Z$ or $\theta =\frac{2n\mathrm{\pi }}{5},n\in Z$ or $\theta =(2n+1)\pi ,n\in Z$


(viii) $\sin 3\theta -\sin \theta =4{\cos }^2\theta -2$
$$\begin{gathered} \Rightarrow \sin 3\theta -\sin \theta =2(2{\cos }^2\theta -1) \\ \Rightarrow 2\sin \left(\frac{2\theta }{2}\right)\cos \left(\frac{4\theta }{2}\right)=2\cos 2\theta \\ \Rightarrow 2\sin \theta \cos 2\theta =2\cos 2\theta \\ \Rightarrow \sin \theta \cos 2\theta =\cos 2\theta \\ \Rightarrow \cos 2\theta (\sin \theta -1)=0 \end{gathered}$$
$\Rightarrow \cos 2\theta =0$ or $\sin \theta -1=0$
$\Rightarrow$$\cos 2\theta =\cos \frac{\mathrm{\pi }}{2}$ or $\sin \theta =1\Rightarrow \sin \theta =\sin \frac{\mathrm{\pi }}{2}$
$\Rightarrow 2\theta =(2n+1)\frac{\mathrm{\pi }}{2}$, $n\in Z$ or $\theta =n\pi +(-1{)}^n\frac{\mathrm{\pi }}{2},n\in Z$
$\Rightarrow \theta =(2n\hspace{0.17em}+1)\frac{\mathrm{\pi }}{4},n\in Z$ or $\theta =n\pi +(-1{)}^n\frac{\mathrm{\pi }}{2},n\in Z$


(ix) $\sin 2\theta -\sin 4\theta +\sin 6\theta =0.$
$$\begin{gathered} \Rightarrow 2\sin \left(\frac{8\theta }{2}\right)\cos \left(\frac{4\theta }{2}\right)-\sin 4\theta =0 \\ \Rightarrow 2\sin 4\theta \cos 2\theta -\sin 4\theta =0 \\ \Rightarrow \sin 4\theta (2\cos 2\theta -1)=0 \end{gathered}$$
$\Rightarrow \sin 4\theta =0$ or $2\cos 2\theta -1=0$
$\Rightarrow 4\theta =n\pi$, $n\in Z$ or $\cos 2\theta =\frac{1}{2}\Rightarrow \cos 2\theta =\cos \frac{\mathrm{\pi }}{3}$
$\Rightarrow \theta =\frac{n\mathrm{\pi }}{4},n\in Z$ or $\theta =n\pi \pm \frac{\mathrm{\pi }}{6},n\in Z$