Question

What is the polar form of `( -4,5 )`?

Answer 1

The polar form of (-4,5) has `sqrt(41)` as module and `arccos(-4/sqrt(41))` as argument.

Explanation:

You can use Pythagoras theorem or the complex numbers. I'm gonna use the complex numbers because it is simpler to write down and to explain as I always do that and english is not my mother language.

By identifying `RR^2` as the complex plan `CC`, `(-4,5)` is the complex number `-4 + 5i`. Its module is `abs(-4+5i) = sqrt(5^2 + (-4)^2) = sqrt(41)`.

We now need the argument of this complex number. We know its module, so we can write that `-4+5i = sqrt41(-4/sqrt41 + i5/sqrt41)`.

We know that when we factorize by the module, we get the cosine and the sine of a real number. It means that `EE alpha in RR` such that `cos(alpha) = -4/sqrt41` and `sin(alpha) = 5/sqrt(41)`. So `alpha = arccos(-4/sqrt(41))` which is the argument of (-4,5).