The answer can be found using the conversion of 1 radian equals $\displaystyle \frac{180}{\pi}$ degrees.  Multiplying $\displaystyle \frac{3\pi}{2}$ by this conversion factor gives 270 degrees.

Set $\displaystyle x=1, y=3$. Calculate the polar coordinates $\displaystyle (r,\theta )$ as follows:

$\displaystyle r = \sqrt{x^{2}+y^{2}}= \sqrt{1^{2}+3^{2}}=\sqrt{10} \approx 3.16$

$\displaystyle \theta = \arctan \frac{y}{x}= \arctan \frac{3}{1} = \arctan 3 \approx 1.25$

$\displaystyle \frac{x^\circ}{\frac{\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply:

$\displaystyle x^\circ*\pi=180^\circ*\frac{\pi}{6}$

Notice that the $\displaystyle \pi$'s cancel out:

$\displaystyle x^\circ=180^\circ*\frac{1}{6}$

$\displaystyle x^\circ=30^\circ$

The relationship between degrees and radians is $\displaystyle 180^\circ=\pi$ radians. Therefore, $\displaystyle 180^\circ$ would be $\displaystyle \pi$ radians.

Since $\displaystyle 180 ^{\circ } = \pi \textrm{ rad}$, we can convert as follows:

$\displaystyle 125 \cdot \frac{\pi }{180} = \frac{25\pi }{36}$

Since there are $\displaystyle \pi$ radians for every $\displaystyle 180^\circ$, we can set up a proportion to solve.

$\displaystyle \frac{85^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply.

$\displaystyle 85^\circ*\pi=180^\circ *x$

Divide both sides by $\displaystyle 180^\circ$.

$\displaystyle \frac{85^\circ}{180^\circ}*\pi=x$

$\displaystyle \frac{17}{36}\pi=x$

The conversion for radians is $\displaystyle 180^\circ=\pi$, so we can make a ratio:

$\displaystyle \frac{270^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$\displaystyle 270^\circ*\pi=180^\circ*x$

Isolate $\displaystyle x$:

$\displaystyle \frac{270^\circ}{180^\circ}\pi=x$

$\displaystyle \frac{3\pi}{2}=x$

The conversion for radians is $\displaystyle 180^\circ=\pi$, so we can make a ratio:

$\displaystyle \frac{x^\circ}{\frac{\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$\displaystyle x^\circ*\pi=180^\circ*\frac{\pi}{3}$

Divide both sides by $\displaystyle \pi$:

$\displaystyle x^\circ=\frac{180^\circ}{3}$

$\displaystyle x^\circ=60^\circ$

The conversion for radians is $\displaystyle 180^\circ=\pi$, so we can make a ratio:

$\displaystyle \frac{x^\circ}{\frac{8\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$\displaystyle x^\circ*\pi=180^\circ*\frac{8\pi}{3}$

Notice that the $\displaystyle \pi$'s cancel out:

$\displaystyle x^\circ=180^\circ*\frac{8}{3}$

$\displaystyle x^\circ=480^\circ$

The conversion for radians is $\displaystyle 180^\circ=\pi$, so we can make a ratio:

$\displaystyle \frac{x^\circ}{\frac{3\pi}{4}}=\frac{180^\circ}{\pi}$

Cross multiply:

$\displaystyle x^\circ*\pi=180^\circ*\frac{3\pi}{4}$

Notice that the $\displaystyle \pi$'s cancel out:

$\displaystyle x^\circ=180^\circ*\frac{3}{4}$

$\displaystyle x^\circ=135^\circ$