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

Oct 25, 2016

The trigonometric form is $2 \sqrt{2} \left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right)$

#### Explanation:

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

If $z = a + i b$ 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 $z = r \left(\cos \theta + i \sin \theta\right)$

and we get $\cos \theta = \frac{1}{\sqrt{2}}$
and $\sin \theta = \frac{1}{\sqrt{2}}$

so, $\theta = \frac{\pi}{4}$
and the trigonometric form is 2sqrt2(cos(pi/4)+isin(pi/4)

And the exponential form is $z = 2 \sqrt{2} {e}^{i \frac{\pi}{4}}$