Algebra and Trigonometry, 5th Edition

Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

To graph the tangent function, we could make a table of values using a calculator, but in this case it is easier to begin with the definition of tangent and the coordinates of a few points on the unit circle. We recall that

$$\text{tan}s={\displaystyle \frac{y}{x}}={\displaystyle \frac{\text{sin}s}{\text{cos}s}}.$$

The tangent function is not defined when x, the first coordinate, is 0. That is, it is not defined for any number s whose cosine is 0:

$$s=\pm {\displaystyle \frac{\pi }{2}},\pm {\displaystyle \frac{3\pi }{2}},\pm {\displaystyle \frac{5\pi }{2}},\dots .$$

We draw vertical asymptotes at these locations (see Fig. 1 below).

Figure 1.