The last three circular functions complete the picture. The cotangent is read on the horizontal axis that touches the cycle at $B$; the secant and cosecant appear when we draw the tangent line to the cycle in $P$ and see where it cuts the axes.

The story

The names “secant” and “tangent” come from the geometry of the circle: the tangent line touches the circumference (tangere, touch) and the secant line cut (secare, cut). In the definition figure, the segment $\overline{OS}$ lies on a straight line that, when extended, is secant to the cycle, and the segment $\overline{AT}$ lies on the tangent line at $A$. These segments were already tabulated by Arab astronomers: Abu al-Wafa, in the 10th century, organized the six functions and showed how each one derives from the others.

Abu al-Wafa (940–998), later artistic representation.
Abu al-Wafa (940–998), later artistic representation.Unidentified author · Wikimedia Commons · Public domain

Sources: Abu al-Wafa (MacTutor) · Trigonometric functions (MacTutor).

In everyday life

  • Roofs (secant). The rafter spanning half the width $d$ of a roof with a slope $\alpha$ measures $d\sec\alpha$; the carpenter cuts the wood for this account.
  • Ramps (cosecant). To climb a height $h$ with inclination $\alpha$, the ramp measures $h\operatorname{cossec}\alpha$. With the maximum slope of the Brazilian accessibility standard (8,33%, $\alpha \approx 4{,}76^\circ$), reaching 50 cm requires around 6 m of ramp.
  • Shadows (cotangent). A stick of height $h$ with the Sun at an angular height $\alpha$ casts a shadow of $h\operatorname{cotg}\alpha$ — the principle of sundials.
The length of a ramp is the height times the cosecant of the slope. Test the 8,33% limit of the accessibility standard.

Theory

cotangent function

DefiniçãoCotangent and cotangent function

Given a real number $x$, with $x \ne k\pi$, let $P$ be its image in the cycle. Consider the straight line $\overleftrightarrow{OP}$ and let $D$ be its intersection with the cotangent axis. The cotangent of $x$ (we indicate $\operatorname{cotg} x$) is the algebraic measure of the segment $\overline{BD}$.

The cotangent function is the function $f\colon D \to \mathbb{R}$ that associates the real $\overline{BD} = \operatorname{cotg} x$ to each real $x \ne k\pi$.

For $x = k\pi$, the point $P$ is in $A$ or $A'$ and the line $\overleftrightarrow{OP}$ is parallel to the cotangent axis: there is no $D$, and $\operatorname{cotg} x$ is not defined.

Drag $P$. The straight line $OP$ cuts the cotangent axis (horizontal, through $B$) at $D$; $\overline{BD} = \operatorname{cotg} x$.

PropriedadeProperties of the cotangent function

  1. The domain is $D = \{x \in \mathbb{R} \mid x \ne k\pi\}$.
  2. The image is $\mathbb{R}$: for every real $y$ there is real $x$ with $\operatorname{cotg} x = y$.
  3. If $x$ is from the 1st or 3rd quadrant, $\operatorname{cotg} x$ is positive.
  4. If $x$ is from the 2nd or 4th quadrant, $\operatorname{cotg} x$ is negative.
  5. If $x$ passes through any of the four quadrants, $\operatorname{cotg} x$ is decreasing.
  6. The cotangent function is periodic and its period is $\pi$.
View the demos

2. Given $y$, mark the point $D$ with $\overline{BD} = y$ on the cotangent axis; the straight line $\overleftrightarrow{OD}$ cuts the cycle at two points, real images whose cotangent is $y$.

3 and 4. In the 1st and 3rd quadrants, the straight line $OP$ cuts the axis $d$ to the right of $B$ ($\overline{BD} > 0$); in the 2nd and 4th, on the left ($\overline{BD} < 0$).

5. In the 1st quadrant, when $x$ increases from $0$ to $\frac{\pi}{2}$, the straight line $OP$ becomes more and more “upright” and the point $D$ approaches $B$ coming from the right: $\overline{BD}$ decreases (from $+\infty$ to 0). In the right-angled triangle $OBD$, with fixed side $OB = 1$, the side $BD$ decreases when the angle $B\hat OD$ decreases, and this angle is the complement of $x$. The other quadrants are analogous.

6. $x$ and $x + k\pi$ have coincident or diametrically opposite images, which determine the same straight line $OP$ and the same point $D$. Therefore $\operatorname{cotg}(x + k\pi) = \operatorname{cotg} x$, and the smallest positive value of $k\pi$ is $\pi$.

Cotangent graph. The asymptotes are the lines $x = k\pi$.

Drying function

DefiniçãoSecant and secant function

Given a real number $x$, with $x \ne \dfrac{\pi}{2} + k\pi$, let $P$ be its image in the cycle. Consider the straight line $s$ tangent to the cycle in $P$ and let $S$ be its intersection with the cosine axis. The secant of $x$ (we indicate $\sec x$) is the abscissa $\overline{OS}$ of the point $S$.

The secant function is the function $f\colon D \to \mathbb{R}$ that associates the real $\overline{OS} = \sec x$ to each real $x \ne \dfrac{\pi}{2} + k\pi$.

For $x = \dfrac{\pi}{2} + k\pi$, the point $P$ is in $B$ or $B'$, the line $s$ is parallel to the cosine axis and $\sec x$ is not defined.

Drag $P$. The tangent line to the cycle at $P$ cuts the cosine axis at $S$ ($\overline{OS} = \sec x$) and the sine axis at $C$ ($\overline{OC} = \operatorname{cossec} x$).

PropriedadeProperties of the secant function

  1. The domain is $D = \left\{x \in \mathbb{R} \;\middle|\; x \ne \dfrac{\pi}{2} + k\pi\right\}$.
  2. The image is $\mathbb{R} - \,]-1, 1[$: for every $y$ with $y \le -1$ or $y \ge 1$ there is $x$ with $\sec x = y$.
  3. If $x$ is from the 1st or 4th quadrant, $\sec x$ is positive.
  4. If $x$ is from the 2nd or 3rd quadrant, $\sec x$ is negative.
  5. If $x$ runs through the 1st or 2nd quadrant, $\sec x$ is increasing.
  6. If $x$ goes through the 3rd or 4th quadrant, $\sec x$ is decreasing.
  7. The secant function is periodic and its period is $2\pi$.
View the demos

The key is the triangle $OPS$: it is rectangle at $P$ (the radius is perpendicular to the tangent) and has a side $OP = 1$. If the angle $P\hat OS$ measures $\theta$, then $\cos\theta = \dfrac{OP}{OS} = \dfrac{1}{OS}$, that is, $|\overline{OS}| = \dfrac{1}{|\cos x|}$; and $S$ is on the same side of $O$ as the projection of $P$ on the $u$ axis. Therefore

$$\sec x = \frac{1}{\cos x}.$$

2. As $0 < |\cos x| \le 1$, we have $|\sec x| \ge 1$. Conversely, if $|y| \ge 1$, then $\frac{1}{y} \in [-1, 1]$ and there is $x$ with $\cos x = \frac{1}{y}$, that is, $\sec x = y$.

3 and 4. $\sec x$ has the sign of $\cos x$: positive in the 1st and 4th quadrants, negative in the 2nd and 3rd.

5 and 6. In the 1st quadrant, $\cos x$ decreases and is positive, therefore $\frac{1}{\cos x}$ increases; in the 2nd, $\cos x$ decreases and is negative, so the opposite also increases (it goes from $-\infty$ to $-1$). In the 3rd and 4th, $\cos x$ increases and the opposite decreases.

7. $\cos(x + 2k\pi) = \cos x$ implies $\sec(x + 2k\pi) = \sec x$; the smallest positive period is $2\pi$, as $\sec 0 = 1$ only repeats after a full revolution.

Secant graph. The arcs are above $y = 1$ or below $y = -1$; the asymptotes are $x = \frac{\pi}{2} + k\pi$.

Cosecant function

DefiniçãoCosecant and cosecant function

Given a real number $x$, with $x \ne k\pi$, let $P$ be its image in the cycle. Consider the straight line $s$ tangent to the cycle in $P$ and let $C$ be its intersection with the sine axis. The cosecant of $x$ (we indicate $\operatorname{cossec} x$) is the ordinate $\overline{OC}$ of the point $C$.

The cosecant function is the function $f\colon D \to \mathbb{R}$ that associates the real $\overline{OC} = \operatorname{cossec} x$ to each real $x \ne k\pi$.

For $x = k\pi$, the point $P$ is in $A$ or $A'$, the line $s$ is parallel to the sine axis and $\operatorname{cossec} x$ is not defined.

PropriedadeProperties of the cosecant function

  1. The domain is $D = \{x \in \mathbb{R} \mid x \ne k\pi\}$.
  2. The image is $\mathbb{R} - \,]-1, 1[$.
  3. If $x$ is from the 1st or 2nd quadrant, $\operatorname{cossec} x$ is positive.
  4. If $x$ is from the 3rd or 4th quadrant, $\operatorname{cossec} x$ is negative.
  5. If $x$ runs through the 2nd or 3rd quadrant, $\operatorname{cossec} x$ is increasing.
  6. If $x$ passes through the 1st or 4th quadrant, $\operatorname{cossec} x$ is decreasing.
  7. The cosecant function is periodic and its period is $2\pi$.
View the demos

In the triangle $OPC$, a rectangle in $P$ with $OP = 1$, the angle $P\hat OC$ is the complement of the angle that $OP$ makes with the axis $u$, and we obtain in the same way

$$\operatorname{cossec} x = \frac{1}{\operatorname{sen} x}.$$

All properties follow from those of the sine: $|\operatorname{sen} x| \le 1$ gives $|\operatorname{cossec} x| \ge 1$; the sign is that of the sine (positive in the 1st and 2nd quadrants); where the sine increases while maintaining the sign (1st and 4th quadrants), the inverse decreases, and where the sine decreases (2nd and 3rd), the inverse increases; and the period is that of the sine, $2\pi$.

Cosecant graph; the highlighted range, from $0$ to $2\pi$, is a complete period. The asymptotes are $x = k\pi$.

ResumoThe six circular functions

function domain image period positive in the quadrants
$\operatorname{sen}$ $\mathbb{R}$ $[-1, 1]$ $2\pi$ 1º e 2º
$\cos$ $\mathbb{R}$ $[-1, 1]$ $2\pi$ 1º e 4º
$\operatorname{tg}$ $x \ne \frac{\pi}{2} + k\pi$ $\mathbb{R}$ $\pi$ 1º e 3º
$\operatorname{cotg}$ $x \ne k\pi$ $\mathbb{R}$ $\pi$ 1º e 3º
$\sec$ $x \ne \frac{\pi}{2} + k\pi$ $\mathbb{R} - \,]-1, 1[$ $2\pi$ 1º e 4º
$\operatorname{cossec}$ $x \ne k\pi$ $\mathbb{R} - \,]-1, 1[$ $2\pi$ 1º e 2º