Secant, cosecant and cotangentEdexcel International A Level Maths: Revision notes

The reciprocals of the three main trigonometric functions have their own names: $\sec x=\frac{1}{\cos x},\qquad \operatorname{cosec}\,x=\frac{1}{\sin x},\qquad \cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}.$ Secant is the reciprocal of cosine, cosecant is the reciprocal of sine and cotangent is the reciprocal of tangent. Each is undefined wherever its denominator is zero: $\sec x$ where $\cos x=0$, and $\operatorname{cosec}\,x$ and $\cot x$ where $\sin x=0$.

Every value follows from sine, cosine and tangent, so use the reciprocal. For an acute angle with $\sin\theta=\frac{5}{13}$: $\cos\theta=\frac{12}{13}$, so $\operatorname{cosec}\,\theta=\frac{13}{5}$, $\sec\theta=\frac{13}{12}$ and $\cot\theta=\frac{12}{5}$. Exact values come the same way: $\sec60^{\circ}=2$, $\operatorname{cosec}\,30^{\circ}=2$, $\sec45^{\circ}=\sqrt2$, $\cot45^{\circ}=1$, $\cot30^{\circ}=\sqrt3$ and $\cot60^{\circ}=\frac{1}{\sqrt3}$. Because $|\sin x|\le1$ and $|\cos x|\le1$, the reciprocals satisfy $|\sec x|\ge1$ and $|\operatorname{cosec}\,x|\ge1$ wherever they are defined.

$y=\sec x$ has period $2\pi$ and vertical asymptotes at $x=\frac{\pi}{2}+n\pi$. It has U-shaped branches: minimum points $(2n\pi,\,1)$ lie above the $x$-axis and maximum points $((2n+1)\pi,\,-1)$ lie below it, so there are no values between $-1$ and $1$. $y=\operatorname{cosec}\,x$ has period $2\pi$ and asymptotes at $x=n\pi$. Its branches turn at $(\frac{\pi}{2},\,1)$ and $(\frac{3\pi}{2},\,-1)$. $y=\cot x$ has period $\pi$ and asymptotes at $x=n\pi$. It crosses the $x$-axis at $x=\frac{\pi}{2}+n\pi$ and decreases between each pair of asymptotes, so it has no turning points. Its range is all real numbers. Each graph sits on a graph you already know: a reciprocal is large where the original is near zero, and equals the original ($\pm1$) where the original is $\pm1$.

None of the three functions is one-to-one over its whole domain, so an inverse needs a restricted domain. The usual choices are:

• $\sec x$ on $0\le x\le\pi$, $x\ne\frac{\pi}{2}$, with range $y\le-1$ or $y\ge1$;
• $\operatorname{cosec}\,x$ on $-\frac{\pi}{2}\le x\le\frac{\pi}{2}$, $x\ne0$, with range $y\le-1$ or $y\ge1$;
• $\cot x$ on $0<x<\pi$, with range all real numbers. Over these intervals each function takes every value in its range exactly once.

Everything above holds whether angles are in degrees or radians; the period $2\pi$ becomes $360^{\circ}$ and $\pi$ becomes $180^{\circ}$. Match the unit to the question's interval. To solve an equation with a reciprocal function, convert it to sine, cosine or tangent first. Example. Solve $\sec\theta=-2$ for $0\le\theta<360^{\circ}$. $\cos\theta=-\frac12$, so $\theta=120^{\circ}$ or $240^{\circ}$. Example. Solve $\sec\theta=3\cos\theta$. Then $\cos^2\theta=\frac13$, so $\cos\theta=\pm\frac{1}{\sqrt3}$ and there are four solutions: $54.7^{\circ},125.3^{\circ},234.7^{\circ},305.3^{\circ}$.