Graphs of Tangent and Cotangent

Definition For a real number $x$ with $\cos x \ne 0$,

Tangent is undefined where $\cos x = 0$, that is at $x = \frac{\pi}{2} + k\pi$ for every integer $k$. If $P(x) = (\cos x, \sin x)$ is the point of the unit circle that belongs to $x$, then $\tan x$ is the slope of the line through the origin and $P(x)$. That line meets the vertical line tangent to the circle at $(1, 0)$ in the point $(1, \tan x)$, which is where the name comes from.

Definition The tangent graph is the graph of $y = \tan x$ with $x$ in radians. Its domain is all real numbers except $\frac{\pi}{2} + k\pi$ for integers $k$, and its range is all real numbers. It is zero exactly at $x = k\pi$, and it is odd, $\tan(-x) = -\tan x$. On each open interval

the function is increasing and takes every real value exactly once. The piece of the graph over one of these intervals is called a branch.

Definition The line $x = a$ is a vertical asymptote of the graph of $f$ if $f(x)$ increases without bound or decreases without bound as $x$ approaches $a$ from at least one side, meaning

The tangent graph has a vertical asymptote at every $x = \frac{\pi}{2} + k\pi$. Near such a point $\sin x$ is close to $\pm 1$ while $\cos x$ is close to $0$, so the quotient is very large in size. At $x = \frac{\pi}{2}$ the cosine is positive just to the left and negative just to the right, so $\tan x \to \infty$ as $x \to \frac{\pi}{2}^-$ and $\tan x \to -\infty$ as $x \to \frac{\pi}{2}^+$.

Definition The tangent function has period $\pi$. For every $x$ in its domain, $x + \pi$ is also in the domain and

and no smaller positive number has this property. More generally, for constants $A \ne 0$ and $B \ne 0$ the function $y = A\tan\bigl(B(x - C)\bigr) + D$ has period $\frac{\pi}{\lvert B \rvert}$.

Definition For a real number $x$ with $\sin x \ne 0$,

Cotangent is undefined at $x = k\pi$ for every integer $k$. Wherever $\tan x$ is defined and not zero, $\cot x = \frac{1}{\tan x}$. At $x = \frac{\pi}{2} + k\pi$ tangent is undefined, but cotangent is defined there and equals $0$.

Definition The cotangent graph is the graph of $y = \cot x$ with $x$ in radians. Its domain is all real numbers except $k\pi$ for integers $k$, its range is all real numbers, and it has period $\pi$. It is odd, it is zero exactly at $x = \frac{\pi}{2} + k\pi$, and its vertical asymptotes are the lines $x = k\pi$. On each interval $\bigl(k\pi, (k + 1)\pi\bigr)$ it is decreasing and takes every real value exactly once. For every $x$ that is not a multiple of $\pi$

so the cotangent graph is the tangent graph shifted $\frac{\pi}{2}$ units to the right and then reflected across the $x$-axis. Shifting and reflecting do not change the period, which is why cotangent repeats every $\pi$ units just like tangent.