Unit circle

In mathematics, the unit circle of the circle whose radius is the length 1 and the center of which coincides with the coordinate origin of a Cartesian coordinate system of the plane.

Trigonometric relationships

There is a point on the unit circle, one can define an angle with the x-axis ( abscissa), is seen from below the origin of the coordinate system of. Then for the coordinates of

With the aid of relations in a right triangle, the following relationships can be established:

The oriented length of the tangent to the circle, which is perpendicular to the x- axis to the apex of the angle of the tangent.

The unit circle can be represented using the Euler's identity:

Rational parametrization

Even without resorting to trigonometric functions allow all points of the unit circle, see. Be an arbitrary real number. An intersection of the straight through and with the unit circle is trivially. The other is with him, and goes through when all goes through the whole circle. The point is, however, this is only possible after the border crossing.

This parameterization is suitable for the body. For rational is obtained from it by elementary transformations Pythagorean triples.

Other standards

If a different standard used as the Euclidean norm 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 a rhombus whose vertices lie on the coordinate axes at a distance one to the origin.