Question

How do you convert the Cartesian coordinates (10,10) to polar coordinates?

Answer 1

Cartesian: `(10;10)`

Polar: `(10sqrt2;pi/4)`

Explanation:

The problem is represented by the graph below:

In a 2D space, a point is found with two coordinates:

The cartesian coordinates are vertical and horizontal positions `(x;y)`.

The polar coordinates are distance from origin and inclination with horizontal `(R,alpha)`.

The three vectors `vecx, vecy and vecR` create a right triangle in which you can apply the pythagorean theorem and the trigonometric properties. Thus, you find:

`R=sqrt(x^2+y^2)`

`alpha=cos^(-1)(x/R)=sin^(-1)(y/R)`

In your case, that is:

`R=sqrt(10^2+10^2)=sqrt(100+100)=sqrt200=10sqrt2`

`alpha=sin^(-1)(10/(10sqrt2))=sin^(-1)(1/sqrt2)=45°=pi/4`