Question

How do you write the trigonometric form of `2+2i`?

Answer 1

The trigonometric form is `2sqrt2(cos(pi/4)+isin(pi/4))`

Explanation:

Let `z=2+2i`
To calculate the trigonomrtric version, we need to calculate the modulus of the complex number.

If `z=a+ib` then the modulus is `∣z∣=sqrt(a^2+b^2)`

So here `∣z∣=sqrt(2^2+2^2)=2sqrt2`
Then `z/(∣z∣)=1/sqrt2+(i)/sqrt2`

tHen we compare this to `z=r(costheta+isintheta)`

and we get `costheta=1/sqrt2`
and `sintheta=1/sqrt2`

so, `theta=pi/4`
and the trigonometric form is `2sqrt2(cos(pi/4)+isin(pi/4)`

And the exponential form is `z=2sqrt2e^(ipi/4)`