Three quotients, three domains

The reciprocal functions expand the circular family:

$$\cot x=\frac{\cos x}{\sin x},\qquad \sec x=\frac1{\cos x},\qquad\csc x=\frac1{\sin x}.$$
Function Points excluded Image Fundamental period
$\cot x$ $x=k\pi$ $\mathbb R$ $\pi$
$\sec x$ $x=\pi/2+k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$
$\csc x$ $x=k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$

In the entire table, $k\in\mathbb Z$. The cotangent is not defined by $1/\tan x$ in its entire domain: in $x=\pi/2$, the cotangent is worth 0, while the tangent does not exist. The identity $\cot x=1/\tan x$ requires that both expressions exist.

Demonstrations by definitions

Like $|\cos x|\le1$ and it is not zero in the secant domain, $|1/\cos x|\ge1$. This proves the restriction of the secant image. The same argument applies to the cosecant. The permitted values ​​are all reached because sine and cosine run through $[-1,1]$.

Using sine and cosine parities:

$$\cot(-x)=-\cot x,\quad\sec(-x)=\sec x,\quad\csc(-x)=-\csc x.$$

Cotangent and cosecant are odd; secant is even. A half-turn preserves the quotient of the cotangent, but changes the sign of the two reciprocals. Therefore, its fundamental periods are those presented in the table.

How to sketch the graphs

The cotangent decreases from unbounded positive values ​​to unbounded negative values ​​in each branch $(k\pi,(k+1)\pi)$; resets to $\pi/2+k\pi$.

For the secant, use the cosine graph: where it is worth 1 or $-1$, the reciprocal has the same value. Where the cosine approaches zero, the secant increases in magnitude. Its branches fall outside the range $-1

The cosecant is constructed in the same way from the sine. Its vertical asymptotes are the zeros of the sine; those of the secant are the zeros of the cosine.

A complete calculation

In $x=\pi/6$, $\sin x=1/2$ and $\cos x=\sqrt3/2$. Then:

$$\cot x=\sqrt3,\quad\sec x=\frac{2\sqrt3}{3},\quad\csc x=2.$$

In everyday life

  • Roofs. The rafter of a roof with pitch $\alpha$ and half width $d$ measures $d\sec\alpha$.
  • Ramps. The length of a ramp that goes up $h$ with slope $\alpha$ is $h\operatorname{cossec}\alpha$.
  • Shadows. A stick of height $h$ casts a shadow $h\operatorname{cotg}\alpha$ when the Sun is at a height $\alpha$.
Ramp length: height times cosecant of the slope.

Exercises with resolution

1. Calculate $\cot(\pi/2)$, $\sec\pi$ and $\csc(3\pi/2)$.

Check resolution 1

The values ​​are, respectively, $0$, $-1$ and $-1$.

2. Does equation $\sec x=1/2$ have a real solution?

Check resolution 2

No: in the secant domain, its modulus is at least 1.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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