Topic 3.11 The Secant, Cosecant, and Cotangent Functions: define the reciprocal trigonometric functions and identify their periods, asymptotes, ranges and graphs.

What this topic is asking

The College Board (Topic 3.11) wants you to define the three reciprocal trigonometric functions: secant, cosecant and cotangent. Each is the reciprocal of one of the primary functions, so each has vertical asymptotes where its underlying function is zero, and a range that excludes the interval near the midline.

The reciprocal definitions

A memory aid: the "co" pairing is crossed, secant pairs with cosine, cosecant with sine. Reading each as one-over-its-primary makes every feature follow from the primary function.

Asymptotes from the denominators

Where the primary function reaches $\pm 1$ (its peaks and troughs), the reciprocal reaches $\pm 1$ too, marking the turning points of each U-shaped branch.

Periods and ranges

How the graphs relate

A point worth stating once is that each reciprocal graph is a "mirror through $y = 1$ and $y = -1$" of its primary. Where sine or cosine is large, the reciprocal is small (down to $\pm 1$); where sine or cosine approaches zero, the reciprocal shoots off to $\pm\infty$, creating the asymptote. The U-shaped branches of secant and cosecant tuck exactly into the peaks and troughs of cosine and sine, touching them at $\pm 1$. Seeing the reciprocal as inheriting every zero of its primary as an asymptote, and every extreme as a turning point, lets you sketch all three from the sine and cosine graphs you already know.

Try this

Q1. Where does $y = \cot x$ have vertical asymptotes? [1 point]

• Cue. Where $\sin x = 0$, that is at $x = \pi k$, since $\cot x = \frac{\cos x}{\sin x}$.

Q2. What is the period of $y = \cot x$? [1 point]

• Cue. $\pi$, the same as tangent, since $\cot x = \frac{1}{\tan x}$.