The tangent measures the slope of the straight line that connects the center of the cycle to the point $P$. It is read on a vertical axis that touches the cycle at $A$ — the tangent axis — and, unlike sine and cosine, it can assume any real value.

The story

The tangent was born far from the circle: it was born from shadows. A vertical stick stuck into the ground, the gnomon, casts a shadow whose length depends on the height of the Sun — and the ratio between the shadow and the stick is, in today’s language, a tangent (or a cotangent, depending on the position of the stick). Because of sundials and prayer times, astronomers in the Islamic world needed these ratios, and the first known shadow tables were made in Baghdad around 860. They are usually attributed to Habash al-Hasib. A century and a half later, al-Biruni brought together everything that was known in the Exhaustive Treatise on Shadows. The name “tangent” would only appear in 1583, with Thomas Fincke (see chapter VIII).

The two shadows of the Arab astronomers. The standing stick projects a straight shadow on the ground, which is the cotangent of the Sun’s height; the stick stuck in a wall projects the verse shadow onto the wall, which is the tangent.

Sources: Trigonometric functions (MacTutor) · al-Biruni (MacTutor).

In everyday life

  • Ramps and roofs. The slope of a ramp, in percentage, is $100 \cdot \operatorname{tg}\alpha$: a 8% ramp rises 8 cm for each horizontal meter.
  • Shadows. A rod of height $h$ illuminated by the Sun at an angular height $\alpha$ casts a shadow of length $\dfrac{h}{\operatorname{tg}\alpha}$.
The slope of a ramp in percentage is 100 times the tangent of the angle.

Theory

Definition

DefiniçãoTangent and tangent 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 $\overleftrightarrow{OP}$ and let $T$ be its intersection with the tangent axis. The tangent of $x$ (we indicate $\operatorname{tg} x$) is the algebraic measure of the segment $\overline{AT}$.

The tangent function is the function $f\colon D \to \mathbb{R}$ that associates each real $x \ne \dfrac{\pi}{2} + k\pi$ with the real $\overline{AT} = \operatorname{tg} x$, that is, $f(x) = \operatorname{tg} x$.

For $x = \dfrac{\pi}{2} + k\pi$, the point $P$ is in $B$ or $B'$, and the line $\overleftrightarrow{OP}$ is parallel to the tangent axis. As there is no point $T$, $\operatorname{tg} x$ is not defined.

Drag $P$. The straight line $OP$ cuts the tangent axis at $T$; the algebraic measure of $\overline{AT}$ is $\operatorname{tg} x$. Near $B$ and $B'$, the point $T$ runs away to infinity.

Properties

PropriedadeProperties of the tangent function

  1. The domain of the tangent function is $D = \left\{x \in \mathbb{R} \;\middle|\; x \ne \dfrac{\pi}{2} + k\pi\right\}$.
  2. The range of the tangent function is $\mathbb{R}$: for every real $y$ there is a real $x$ with $\operatorname{tg} x = y$.
  3. If $x$ is from the 1st or 3rd quadrant, $\operatorname{tg} x$ is positive.
  4. If $x$ is from the 2nd or 4th quadrant, $\operatorname{tg} x$ is negative.
  5. If $x$ runs through any of the four quadrants, $\operatorname{tg} x$ is increasing.
  6. The tangent function is periodic and its period is $\pi$.
View the demos

2. Given $y \in \mathbb{R}$, consider on the tangent axis the point $T$ such that $\overline{AT} = y$. The straight line $\overleftrightarrow{OT}$ intersects the cycle at two points, $P$ and $P'$, images of real numbers $x$ whose tangent is $y$.

3 and 4. In the 1st and 3rd quadrants, the straight line $OP$ cuts the axis of the tangents above $A$, and $\overline{AT}$ is positive; in the 2nd and 4th, it cuts below $A$, and $\overline{AT}$ is negative.

5. Let us prove, for example, when $x$ runs through the 1st quadrant. Given $x_1 < x_2$ in this quadrant, the central angles satisfy $\alpha_1 < \alpha_2$. The straight lines $OP_1$ and $OP_2$ cut the tangent axis at $T_1$ and $T_2$, and, in the right-angled triangle $OAT$ with fixed side $OA = 1$, the opposite side grows with the angle at $O$: $\overline{AT_1} < \overline{AT_2}$, that is, $\operatorname{tg} x_1 < \operatorname{tg} x_2$. The other quadrants are analogous.

Ative o JavaScript para ver esta figura interativa.

6. If $\operatorname{tg} x = \overline{AT}$ and $k \in \mathbb{Z}$, then $\operatorname{tg}(x + k\pi) = \overline{AT}$, since $x$ and $x + k\pi$ have images $P$ and $P'$ coincident or diametrically opposite in the cycle; thus $\overleftrightarrow{OP} = \overleftrightarrow{OP'}$ and therefore $\overleftrightarrow{OP} \cap c = \overleftrightarrow{OP'} \cap c$. Then, for all $x \ne \frac{\pi}{2} + k\pi$,

$$\operatorname{tg} x = \operatorname{tg}(x + k\pi),$$

and the tangent function is periodic. The period is the smallest positive value of $k\pi$, that is, $\pi$.

Graph: the tangentoid

When $x$ goes through $[0, 2\pi]$ and $P$ makes a complete turn counterclockwise, the algebraic measure $\overline{AT}$ varies like this:

$x$ $0$ $\frac{\pi}{2}$ $\pi$ $\frac{3\pi}{2}$ $2\pi$
$\operatorname{tg} x$ $0$ grows $\nexists$ grows $0$ grows $\nexists$ grows $0$

The graph of $f(x) = \operatorname{tg} x$ is called tangentoid:

The tangentoid. The highlighted range, between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, contains a period; the dashed lines $x = \frac{\pi}{2} + k\pi$ are the asymptotes, where the tangent does not exist.
The point $T$ on the tangent axis and the graph. In $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ the tangent does not exist.