Unit circle

Points on the unit circle ${\displaystyle (\cos \varphi ,\sin \varphi )}$

In mathematics , the unit circle is the circle whose radius has the length 1 and whose center point corresponds to the origin of a Cartesian coordinate system of the plane. The unit circle therefore consists of the points on the plane for which applies. ${\displaystyle (x,y)}$ ${\displaystyle x^{2}+y^{2}=1}$

Trigonometric relationships

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 ${\displaystyle P}$ ${\displaystyle \varphi }$ ${\displaystyle P}$ ${\displaystyle (x_{p},y_{p})}$ ${\displaystyle P}$

${\displaystyle x_{p}=\cos \varphi }$, and ${\displaystyle y_{p}=\sin \varphi }$ ${\displaystyle y_{p}/x_{p}=\tan \varphi .}$

With the help of the relationships in the right triangle , the following relationships can be established:

${\displaystyle \sin \varphi ={\frac {\text{Gegenkathete}}{\text{Hypotenuse}}}}$

${\displaystyle \cos \varphi ={\frac {\text{Ankathete}}{\text{Hypotenuse}}}}$

${\displaystyle \tan \varphi ={\frac {\text{Gegenkathete}}{\text{Ankathete}}}}$

${\displaystyle \cot \varphi ={\frac {\text{Ankathete}}{\text{Gegenkathete}}}}$

In addition, there are the less common functions secant and cosecant , which are defined as the reciprocal functions of cosine and sine.

The oriented length of the tangent to the circle, which is perpendicular to the x-axis, up to the vertex of the angle is the tangent of . ${\displaystyle \varphi }$

The unit circle can also be represented by Euler's identity :

${\displaystyle e^{i\varphi }=\cos \left(\varphi \right)+i\sin \left(\varphi \right)}$.

Rational parameterization

Rational parameterization

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 . ${\displaystyle t}$ ${\displaystyle (-1,0)}$ ${\displaystyle (0,t)}$ ${\displaystyle (-1,0)}$ ${\displaystyle \left({\tfrac {1-t^{2}}{1+t^{2}}},{\tfrac {2t}{1+t^{2}}}\right)}$ ${\displaystyle t}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (-1,0)}$ ${\displaystyle t\to \pm \infty }$

This parameterization is suitable for all bodies. For rational ones one obtains from it Pythagorean triples by elementary transformations . ${\displaystyle t=p/q}$ ${\displaystyle (q^{2}-p^{2},2pq,q^{2}+p^{2})}$

Other norms

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 . ${\displaystyle (\pm 1,\pm 1)}$ ${\displaystyle (\pm 1,0)}$ ${\displaystyle (0,\pm 1)}$

Web links

Wikibooks: Trigonometry (school mathematics)  - learning and teaching materials

Commons : Unit circles  - collection of images, videos and audio files

Wiktionary: unit circle  - explanations of meanings, word origins, synonyms, translations

• Introduction to sine and cosine on the unit circle (video)