Consider a central angle on a unit circle in the cartesian plane with one ray the positive horizontal axis. Use its radian measure, $t$, as the length of the arc measured from the point $(1,0)$.

If $t >0$ the central angle is measured counterclockwise from ray of the positive horizontal axis.

If $t <0$ the central angle is measured clockwise from ray of the positive horizontal axis.

The endpoint of this arc on the unit circle, $P(t) = (x(t), y(t)$, is a function of the number $t$ and the angle measured by $t$.

This function $P: t \to P(t)= (x(t),y(t))$ is called the wrapping function. The wrapping function is visualized by a mapping diagram from a number line to the unit circle in a cartesian plane.

After determining the point on the unit circle $P(t)$ by the wrapping function, a projection function based on the coordinates of $P(t)$ onto a line in that plane is used to determine the sine, cosine, and tangent function values as follows:

Notes: The tangent function is not defined when $x(t) =0$. Thus the domain for the tangent function is $Dom_{tan}=\{ t \in \mathbb R : t \ne (2k+1)\frac\pi 2, k \in \mathbb Z\}$

For $t \in Dom_{tan}$, we have the identity, $tan(t)=\frac{sin(t)}{cos(t)}$.

Notes: The cotangent function is not defined when $y(t) =0$. Thus the domain for the tangent function is $Dom_{cot}=\{ t \in \mathbb R : t \ne k\pi , k \in \mathbb Z\}$

For $t \in Dom_{cot}$, we have the identity, $cot(t)=\frac{cos(t)}{sin(t)}$.