How to find X,Y coordinate given an angle and distance from the origin?

Question

Lets say, the distance from origin is 2.8284271247462 and angle is 45 degrees from North (i.e. +ve y axis) in anticlockwise direction

so this means that answer should be -2,2 , because according to pythagoras that is the answer, if a = 2 and b =2 then c = 2.8284271247462, simple pythagoras.

Is there a way to find X,Y from Given Angle and Length ?

More examples:

if Angle is 315, and distance is 2.8284271247462, then new X,Y = (2,2)
if Angle is 135, and distance is 2.8284271247462, then new X,Y = (-2,-2)
if Angle is 225, and distance is 2.8284271247462, then new X,Y = (2,-2)

Diagram: Diagram

Answer 1

If $r$ is distance from the origin and $\theta$ is angle, then $$ x=r\cos \theta $$ and $$ y=r \sin \theta. $$

Answer 2

Let $X=\mathbb{R}^2$ with the standart topology, given a point $A=(x,y)\in X$ you can write it in terms of the distance from the origen $O=(0,0)$ to the point $A$ namely $r$ and the angle counterclockwise with the $x$ axis namely $\theta$.

The relation is the following: $$x=r\cos \theta , \, y=r\sin \theta$$

where $r=\sqrt{x^2+y^2}$ and $\theta=\arctan\left( \frac{y}{x}\right)$ you can verify it drawing a point $A$ in the plane, identify the lenght form $O$ to $A$ and drawing the perpendicular segment to the $x$ axis that pass through by $A$ and look the triangle that is formed and apply the Pythagorean theorem to see the relations.

Given a explicit $\theta$ and $r$ you can get the point $A=(x,y)$ only making the substitution in the above expression for $x$ and $y$.