Writing in Cartesian form and converting to polar form Question: Write $\frac{u+v}{w}$ in the form $re^{i\theta}$ when $u$=$1$, $v$=$\sqrt3i$, and $w$=$1+i$I added $u$ and $v$ for the Cartesian form because it is easier to do.After adding $u$ and $v$, I get $\frac{1+\sqrt3i}{1+i}$. Need help converting to polar form please
 A: Once you have $\frac{1+\sqrt 3 i}{1+i}$, use rationalization to convert the denominator into something rational.
$$
\frac{1+\sqrt 3 i}{1+i} = \frac{(1+\sqrt 3 i)(1-i)}{(1+i)(1-i)} = \frac{(1 + \sqrt 3) + i(\sqrt 3-1)}2
$$
Once you have done this, you are basically in the form $x+iy$, so you can find the radius $r^2 = x^2+y^2$, so that you can see that:
$$
r^2 = \frac{(\sqrt 3 - 1)^2}{4} + \frac{(\sqrt 3 + 1)^2}{4} = 2
$$
So that $r = \sqrt 2$.
Similarly, $\arctan \frac yx$ is the formula for the angle, which is $\arctan \frac{\sqrt 3 -1}{\sqrt 3 +1} = 15^\circ$.
Whence, you can write the given complex number as $\sqrt 2 \operatorname{cis}{15^\circ}$.
A: The first step you need to take is convert this complex number into the standard form
$$
a + ib \text{ where } a,b \in \mathbb{R}
$$
which you can do by multiplying your number with $(1-i)/(1-i)$. Taking the norm of this number gives the radius $r$, and all you have to do is find your $\theta$
A: Hint: Write $u+v$ and $w$ in polar form and then divide...
you will get something like
$\frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}} \\\ $
Write this as $\frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$
Can anyone change this to a spoiler please :)?
$\frac{2  (\frac{1}{2} +\frac{\sqrt{3}}{2}i)}{\frac{2}{\sqrt{2}}(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)}=\frac{2 e^{i \frac{\pi}{3}}}{\frac{2}{\sqrt{2}}e^{i \frac{\pi}{4}}}=\sqrt{2} e^{i(\frac{\pi}{3}-\frac{\pi}{4})}$
A: Actually divisions are easiest in polar form.
$$u + v = 1 + \sqrt 3  i = 2 * cis( atan(\sqrt3) ) $$
$$w = 1 + i = \sqrt2 * cis( atan(1) ) $$
$$\frac{u+v}{w} = \sqrt2 * cis( atan(\sqrt3) - atan(1) ) \approx 1.414 * cis(0.261) $$
