# When is tan undefined?

The function y = tan(x) is undefined at all points where cos(x) = 0. This is because the tangent of an angle is defined as the sine of that angle divided by the cosine of that angle. In other words, tan(x) = sin(x)/cos(x).

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).

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