Graphing Tangent and the Reciprocal Trig Functions

Tangent from sine and cosine

On the unit circle, $\tan x = \tfrac{\sin x}{\cos x}$. That one fact sets the shape of the graph:

• Zeros where the numerator $\sin x = 0$: at $x = 0, \pm\pi, \pm 2\pi, \ldots$
• Vertical asymptotes where the denominator $\cos x = 0$: at $x = \pm\tfrac{\pi}{2}, \pm\tfrac{3\pi}{2}, \ldots$, about $\pm 1.57$ and $\pm 4.71$.

• y = tan x

Why tangent repeats every π

Adding $\pi$ to an angle turns to the opposite point of the unit circle, which flips the signs of both coordinates: $\sin(x + \pi) = -\sin x$ and $\cos(x + \pi) = -\cos x$. In the quotient the two minus signs cancel:

$$\tan(x + \pi) = \frac{-\sin x}{-\cos x} = \tan x$$

So tangent repeats after half a turn, not a full one. Tangent is the slope of the line from the origin through the point on the circle, and opposite points lie on the same line. Near $x = \tfrac{\pi}{2}$ that line turns vertical and its slope runs off to infinity, which is the asymptote.

For $y = a\tan(bx)$, the period is $\tfrac{\pi}{b}$, and $a$ stretches the graph vertically.

The reciprocal functions

Three more functions are reciprocals of the first three:

$$\sec x = \frac{1}{\cos x} \qquad \csc x = \frac{1}{\sin x} \qquad \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$$

Each has vertical asymptotes where its denominator is $0$. Where cosine is $1$ or $-1$, secant equals it; where cosine is small, secant is large. So the graph of $y = \sec x$ is a row of U-shaped branches, each touching a peak or trough of the cosine wave.

• y = sec x
• y = cos x

Since $\lvert \cos x \rvert \le 1$, its reciprocal is never between $-1$ and $1$: the range of $\sec x$ is $y \le -1$ or $y \ge 1$. Cosecant is the same picture built on the sine wave, shifted by $\tfrac{\pi}{2}$.

Worked examples

Common mistakes

Practice problems