Question

Find general solutions of the following equations:
(a) $\sin \theta =\frac{1}{2}$
(b) $\cos \left(\frac{3\theta }{2}\right)=0$
(c) $\tan \left(\frac{3\theta }{4}\right)=0$
(d) ${\cos }^{2}2\theta =1$
(e) $\sqrt{3}\sec 2\theta =2$
(f) $\csc \left(\frac{\theta }{2}\right)=-1$

Answer 1

(a) Given equation

$\sin \theta =1/2$

$\sin \theta =\sin \left(\pi /6\right)$

If $\sin x=\sin y$

then general formula is $x=n\pi +(-1{)}^{n}y;=0,\pm 2,\pm \mathrm{3.........}$

using this we get , $\theta =n\pi +(-1{)}^n(\pi /6);n=0,\pm 1,\pm 2,\pm \mathrm{3..........}$

$\theta =n\pi +(-1{)}^n(\pi /6);n=0,\pm 1,\pm 2,\pm 3,.......$

(b) Given equation $\therefore \cos \left(3\theta /2\right)=0$

If $\cos x=0$, then $x=(2n+1)\pi /2$ where $n=0,\pm 1,\pm \mathrm{2......}$

using this we get , $\frac{3\theta }{2}=(2n+1)\frac{\pi }{2}$, where $n=0,\pm 1,\pm 2,....$

$\theta =(2n+1)\pi /3wheren=0,\pm 1,\pm 2,...$

(c) $\tan \left(30/4\right)=0$

If $\tan \theta =\tan \alpha then\theta =n\pi +\alpha ,wheren=0,\pm 1,\pm 2,....$

using this we, get $3\theta /4=n\pi wheren=0,\pm 1,\pm 2,........$

$\theta =4n\pi /3wheren=0,\pm 1,\pm 2,.........$

(d) $\sqrt{3}\sec 2\theta =2$

$\cos 2\theta =\sqrt{3}/2\Rightarrow \cos 2\theta =\cos \pi /6$

using formula if $\cos x=\cos ythenx=2n\pi \pm ywheren=0,\pm 1,\pm \mathrm{2......}$

We get $2\theta =2n\pi \pm \pi /6wheren=0,\pm 1,\pm \mathrm{2.......}$

$\theta =n\pi \pm \pi /12wheren=0,\pm 1,\pm 2,........$

(e) $\cos \left(\theta /2\right)=-1$

$\sin \left(\theta /2\right)=-1\Rightarrow \sin \theta /2=\sin 3\pi /2$

$\theta /2=n\pi +(-1{)}^{n}3\pi /2wheren=0,\pm 1,\pm \mathrm{2.......}$

$\theta =2n\pi +(-1)3\pi wheren=0,\pm 1,\pm \mathrm{2.....}$