I'm given a position-vector P=(-1,4) and I'm supposed to find the polar form of it. $r$ is simple enough with Pythagoras, but I'm a bit confused when using the tangent(sorry if my english is not 100% on the technical terms here).
According to the formula, it's supposed to be $tan^{-1}(4/-1)=tan^{-1}(-4) = -76^o $
At least that's what the calculator tells me, but it's obviously wrong. Adding on $ 180^o$ I get $104^o$ which seems to be correct. But how do I go from $-76^o$ to $104^o$ with a better mathematical explanation than "because it looks right"?
Here is how you determine the angle $\theta$ in polar coordinates for a point $(x,y)\in \Bbb R^2$:
$$ \theta = \cases{\arctan(\tfrac{y}{x}) & if $x \gt 0$ \\ \arctan(\tfrac{y}{x}) + \pi & if $x \lt 0$ and $y \ge 0$ \\ \arctan(\tfrac{y}{x}) - \pi & if $x \lt 0$ and $y \lt 0$ \\ \pi/2 & if $x = 0$ and $y \gt 0$ \\ -\pi/2 & if $x = 0$ and $y \lt 0$ \\ \text{undefined} & if $x = 0$ and $y = 0$.} $$
Note you could have figured out all of these by playing with some points in the plane and interpreting the angle $\theta$ as that measured from the positive $x$-axis.
Like you said, knowing when an answer is "obviously wrong" is a good skill to have, and it can be used to derive this definition of $\theta$.