Other than the [[Sinusoidal Functions and their Graphs|sinusoidal functions]], the graphs of trigonometric functions have fewer applications, in part because they have inconvenient characteristics of being undefined at various points. # Tan The graph of $y = \tan x$ is a beautiful sinuous shape snaking vertically from $-\infty$ to $\infty$ with a period of $\pi$. By adding a coefficient to $x$ you can affect the steepness of the shape as well as its period. Here we see $y=\tan x$ and $y= \tan \frac{1}{2} x$. ![[Tan.png]] By adding a coefficient to the entire function we affect steepness without affecting period. Here we have $y=\tan x$ and $y= \frac{1}{2} \tan x$. ![[Tan Graph 2.png]] Note that because of the fact that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ if we add a vertical shift parameter to the left hand side as we have with other functions, the graph in geogebra gets pretty funky. For this reason, add it to the right hand side when actually graphing: ![[Tan Graph 3.png]] Horizontal phase shift works as expected: ![[Tan Graph 4.png]] ## General form with parameters As before we can write any tan graph as $y - k = a\tan m(x -h)$ With: - $k$ being a vertical shift - $h$ being a horizontal phase shift - $m$ affecting period such that $\frac{m}{pi}$ is the period as well as affecting the shape of the graph. - $a$ affecting steepness and if $a<0$, inverting the shape ### Asymptotes $\tan x$ is discontinuous at $x = \frac{n\pi}{2}, n \in \mathbf{Z}$. At these points, $\cos x$ is zero and $\tan x$ becomes undefined. If you think geometrically, this is when we are attempting to take the gradient of a vertical line in the unit circle. As $\tan x$ approaches these asymptotes from the left, it approaches $\infty$ and as it approaches from the right, it approaches $-\infty$.