Sine, cosine and tangent of an arbitrary angle

The unit circle allows the definitions of sine, cosine and tangent, known in Year Two only for acute angles, to be extended to an arbitrary real angle.

Definition — Sine, cosine and tangent

Given a real number $\alpha$ and the corresponding point $P_\alpha$ on the unit circle:

• the cosine of $\alpha$ is the abscissa of $P_\alpha$: $\ \cos\alpha = x_{P_\alpha}$.
• the sine of $\alpha$ is the ordinate of $P_\alpha$: $\ \sin\alpha = y_{P_\alpha}$.
• the tangent of $\alpha$, when $\cos\alpha\ne 0$, is the ratio $\tan\alpha = \dfrac{\sin\alpha}{\cos\alpha}$.

From this definition some fundamental properties follow immediately.

In the simulation below you can drag the point $P$ along the unit circle and watch how the sine, cosine and tangent of the angle $\alpha$ change.

Drag the point $P$ around the unit circle: $\cos\alpha$ (green), $\sin\alpha$ (red) and $\tan\alpha$ (ochre, on the line $x=1$) update in real time.

Links

Topics: Trigonometry
Concepts: Unit circle · Cosine · Sine · Tangent
Functions: Cosine · Sine · Tangent