From the angle to the point

By fixing a starting point and a direction of travel, every real number $\alpha$ identifies a unique point on the unit circle.

Definition — Angle → point association

We fix as the origin of angles the point $A(1;0)$ on the unit circle, and as the positive direction the anticlockwise one. Given a real number $\alpha$ (interpreted as an angle in radians), we associate with $\alpha$ a unique point $P_\alpha$ on the circle, reached by travelling an arc of length $|\alpha|$ starting from $A$:

• anticlockwise if $\alpha > 0$;
• clockwise if $\alpha < 0$.

For $\alpha$ greater than $2\pi$ one simply makes more than one full turn; for $\alpha$ less than $-2\pi$ the same, in the opposite direction.

To the real number $\alpha$ there corresponds the point $P_\alpha$ on the unit circle: its abscissa is $\cos\alpha$, its ordinate is $\sin\alpha$.

Links

Topics: Trigonometry
Concepts: Angle · Unit circle · Radian