The value of a trigonometric function is undefined when the ratio for the function involves division by zero. The function thus has an asymptote at such a point.

The trigonometric function y = tan(x) is undefined at all points where cos(x) = 0. This is because tan(x) is also defined as sin(x) divided by cos(x); namely, tan(x) = `\frac{sin(x)}{cos(x)}` .

To determine the points where tan(x) is undefined, we solve for the equation cos(x) = 0. This gives us x = `\pm\frac{\pi}{2}, \pm\frac{3\pi}{2},\pm\frac{5\pi}{2}, \ldots ` . Therefore, the function y=tan(x) is undefined at all points x = `\frac{\pi}{2} + k\pi` , where k is an integer.

Other properties of the tangent function are as follows:

1) Its domain consists of all real numbers, except the zeroes of the cosine function: that is, all real numbers except `\frac{\pi}{2} + k\pi` , where k is an integer. Meanwhile, its range consists of all real numbers.

2) The interval between two matching points on its graph (period) is `\pi`.

3) Behavior of the graph: from the points 0 to `\frac{\pi}{2}` , the graph increases from 0 to positive infinity; from the points `\frac{\pi}{2}` to `\pi` , the graph increases from negative infinity to 0; and from `\pi` to `\frac{3\pi}{2}` , the graph increases from 0 to positive infinity. Therefore, from the points `\frac{\pi}{2}` to `\frac{3\pi}{2}` , the graph increases from negative infinity to positive infinity. In other words, the tangent is an increasing function between any two successive vertical asymptotes (please refer to the attached graph).