Question

Represent the complex numbers in polar form
$1+\cos \theta +i\sin \theta$

Answer 1

Given $2=1+\cos \theta +\sin \theta ---\left(1\right)$

Every complire number can be written in polar form as

$z=r(\cos x\times i\sin \alpha )----\left(2\right)$

From $\left(1\right)$

Real point $=1+\cos \theta =1+{\cos }^2\frac{\theta }{2}-{\sin }^2\frac{\theta }{2}\{Using\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta \}$

$\Rightarrow 2{\cos }^2\frac{\theta }{2}$

Imaginary part $=\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\{Using\sin 2\theta =2\sin \theta \cos \theta \}$

$\therefore z=2{\cos }^2\frac{\theta }{2}+2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$

$\Rightarrow 2\cos \frac{\theta }{2}\{\cos \frac{\theta }{2}+\sin \frac{\theta }{2}\}----\left(3\right)$

Comparing $\left(2\right)$ and $\left(3\right)$

we see $r=2\cos \frac{\theta }{2}$ and $x=\frac{\theta }{2}$

$\therefore$ Polar form $=2\cos \frac{\theta }{2}\{\cos \frac{\theta }{2}+i\sin \frac{\theta }{2}\}$