## Tangent

(tan)

The **tangent** of an angle is defined using the right-triangle formed by this angle: it is the ratio of the triangle's opposite side to the adjacent side. Calcute implements it with the *tan* function.

tan(1)
1.55740772465

In the above example, the given angle is in radians.

The triangle involved in the definition of the **tangent** function is commonly shown in the cartesian plane with its adjacent side on the X axis and its angle point located at the origin (0,0). The other end of the triangle's hypotenuse marks a point on the periphery of a circle centered at the origin; the circle's radius is the length of the hypotenuse. If the angle is increased from 0 to 2 pi radians (0 to 360 degrees) then the end point of the hypotenuse traces a full circle. As the angle of the hypotenuse changes, so does its **tangent** value as the shape of the corresponding triangle changes to match.

The **tangent** is undefined at 1/4 and 3/4 of a full circle. At these angles, the triangle geometrically vanishes into a simple line segment on the Y axis. The value of the triangle's adjacent side is zero and division by zero is undefined. For example, *tan* is undefined at pi/2 radian, or equivalently at 90 degrees or at 100 gradiens.

The **tangent** function is periodic. Except for those angles that represent 1/4 and 3/4 of a circle, it is defined for all real numbers. As angle values increase (or decrease) outside the range of [0, 2 pi] radians that represent one full rotation, the value of *tan* for these angles repeats itself periodically with each new rotation.