Question

Express the complex number in trigonometric form. 2 - 2i

Answer 1

Substitute the values of θ=−π4θ=-π4 and |z|=2√2|z|=22.2√2(cos(−π4)+isin(−π4))

Answer 2

Answer:

The trigonometric form is $z=2\sqrt{2}(\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4}))$.

Step-by-step explanation:

To find : Express the complex number in trigonometric form 2 - 2i ?

Solution :

First we find the modulus of complex number

If $z=a+ib$ then the modulus is $|z|=\sqrt{a^2+b^2}$

On comparing, a=2, b=-2

So, $|z|=\sqrt{2^2+(-2)^2}$

$|z|=\sqrt{4+4}$

$|z|=2\sqrt{2}$

Then $\frac{z}{|z|}=\frac{1}{\sqrt{2}}+ \frac{i}{\sqrt{2}}$

In trigonometric form, $z=r(\cos\theta+i\sin\theta)$

We get,  $\cos\theta=\frac{1}{\sqrt2}$  and  $\sin\theta=\frac{1}{\sqrt2}$

Which means  $\theta = \frac{\pi}{4}$

The trigonometric form is $z=2\sqrt{2}(\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4}))$.