Trigonometric functions on the unit circle (animation)
If a point lies on the unit circle , then you can define an angle to the x-axis (abscissa) at which you can see from the origin of the coordinate system. The following then applies to
the coordinates of
With the help of the relationships in the right triangle , the following relationships can be established:
Even without resorting to trigonometric functions , all points of the unit circle can be found. Let be any real number. An intersection of the straight line through and with the unit circle is trivial . The other is at , and, when it runs completely through, goes through the whole circle. The point is only reached after crossing the border .
This parameterization is suitable for all bodies. For rational ones one obtains from it Pythagorean triples by elementary transformations .
If a standard other than the Euclidean standard is used for distance measurement, the shape of the unit circle in the Cartesian coordinate system is different. For example, the unit circle for the maximum norm is a square with the corners and the unit circle for the sum norm is a square with the corners and .