Show that range of tangent function is a set of real numbers.

A tangent function.

For any real number there exists a line with a slope equal to that real number.

$y=mx,m\in R$

The tangent function gives the slope of the line segment going from the center of a unit circle to a point on that circle with respect to the angle it makes to the horizonal axis.

The function localid="1647338906306" $\tan$returns a value of zero when the argument is zero. This is a flat horizontal line. It then increases to a value of 1 when the input is a localid="1647338940448" ${45}^{\circ }$angle localid="1647338912731" $\frac{\mathrm{\pi }}{4}$radians. Then as you said at localid="1647338917087" ${90}^{\circ }$we have a vertical line and the slope is undefined.

We can always find a slope of whatever positive value we would like at some angle before that point. Choose a value, no matter how large, and it matches with some point on the circle. This is an example of a vertical asymptote. The function is continuous on each interval in the domain, so it passes through every real number up to whatever large real number you choose. This large real number is arbitrary.

The function is an odd function so when the argument is negative it simply returns the negative of the corresponding positive argument.

So what we have is a function that takes an angle that isn’t vertical and can return any real number that goes with the slope.

localid="1647338921637" $\tan :R\{x:x=\frac{\mathrm{\pi }}{2}+n\mathrm{\pi },\mathrm{n}\in \mathrm{Z}\}\to \mathrm{R}$