Question

How do you convert `(2, 3pi/4)` into cartesian form?

Answer 1

`-sqrt2 + sqrt2 i`

Explanation:

Currently the coordinate `(2,3pi/4)` is in polar form, which is `(r,theta)` form.

We could use a funky looking graph like this in order to plot that point.

However, if we want to convert the coordinate into a more familiar rectangular or cartesian form, we need to get the coordinate into `a+bi` form.

we can use the formula

`r(costheta + isintheta)`

to get the number into

`rcostheta + rsinthetai`

This is because `rcostheta` is the horizontal component of the magnitude of `r`, and `rsintheta` is the vertical component of `r`.

Plugging in for r and theta, in radians, we get

`-sqrt2 + sqrt2 i`

If we were to graph the point, we would do something like this, but instead of 2, we would plot the x axis at `-sqrt2`, and similarly the y coordinate at `sqrt2`