1. ## Exponential form

Write $-1+i$ in exponential form.

So far I have $\sqrt {2} [ ( - \cos ( \frac { \pi }{4} )+ i \sin ( \frac { \pi }{4}) ]$, but then how do I make it look like $\sqrt {2} e^ { i \frac { \pi }{4} }$? The minus sign in front of the cos is giving me problem here.

Thanks.

2. $re^{i\theta} = r(cos(\theta) + isin(\theta))$ is a definition.
You can find a proof of this on wikipedia.
You should rather use $\theta = \frac{3\pi}{4}$

3. You could always use $\frac{3\pi}{4}$. Your cosine can be positive in sign but turn out negative, as its supposed to, and your sine stays positive regardless.

$\sqrt{2}\left(\cos{\left(\frac{3\pi}{4}\right)} + i\sin{\left(\frac{3\pi}{4}\right)}\right) = \sqrt{2}e^{\frac{3\pi i}{4}}$

4. for the angle, consider a set of coordinate axis with the vertical axis as the imaginary and the horizontal axis as the real part of the number.

The real part of the number is -1 and the imaginary part is 1.

This puts you in the second quadrant of the coordinate axis at an angle of $\frac{3\pi}{4}$

This is the inverse tangent of the imaginary part over the real part, but you should always consider the coordinate plane because a calculator will not know if you mean for the real or the imaginary to be negative and may give you an angle that is 180 degrees off.

In other words, the inverse tangent function on your calculator can be used to get the angle from the horizontal axis but you need to know which quadrant you are in to know if you started from the negative or positive side of the real axis.