The sine is the height of the point that rotates in the cycle. When the point turns, the height rises and falls between $-1$ and $1$, drawing the most famous curve in mathematics: the sinusoid.

The story

For more than a thousand years, the sine was a table number: no one thought about drawing its graph. The first known sinusoid appeared around 1634, in a work by the Frenchman Gilles Personne de Roberval on the cycloid, the curve described by a point on a rolling wheel. To calculate the area under the cycloid, Roberval used an auxiliary curve, which he called the “companion of the cycloid” — and this companion is exactly a sinusoid. It was Euler, a century later, who began to treat the sine as a function of a real number, and only then did the graph gain the central place it has today.

The wheel rolls: the point $P$ on the edge draws the cycloid, and the point $Q$, at the same height as $P$ and vertical to the center, draws Roberval’s companion — the first sinusoid.

Sources: Roberval (MacTutor) · Trigonometric functions (MacTutor).

In everyday life

  • Sound. A tuning fork there vibrates 440 times per second; the air pressure near it is a sinusoid of period $\frac{1}{440}$ s.
  • Alternating current. The socket voltage in Brazil is a sinusoid with 60 cycles per second.
  • Tides and seasons. The height of the tide and the length of the day throughout the year are well approximated by functions like $a + b \operatorname{sen}(cx + d)$.
The outlet voltage is a sinusoid of 60 cycles per second, peaking at about 180 V.

Theory

Definition

DefiniçãoSine and sine function

Given a real number $x$, let $P$ be its image in the cycle. The sine of $x$ (we indicate $\operatorname{sen} x$) is the ordinate $\overline{OP_1}$ of the point $P$ in relation to the system $uOv$.

The sine function is the function $f\colon \mathbb{R} \to \mathbb{R}$ that associates each real $x$ with the real $\overline{OP_1} = \operatorname{sen} x$, that is, $f(x) = \operatorname{sen} x$.

Drag $P$ through the cycle: the red segment $\overline{OP_1}$, on the sine axis, is $\operatorname{sen} x$.

Properties

PropriedadeProperties of the sine function

  1. The range of the sine function is the range $[-1, 1]$: $\;-1 \le \operatorname{sen} x \le 1$ for all real $x$.
  2. If $x$ is from the 1st or 2nd quadrant, $\operatorname{sen} x$ is positive.
  3. If $x$ is from the 3rd or 4th quadrant, $\operatorname{sen} x$ is negative.
  4. If $x$ runs through the 1st or 4th quadrant, $\operatorname{sen} x$ is increasing.
  5. If $x$ goes through the 2nd or 3rd quadrant, $\operatorname{sen} x$ is decreasing.
  6. The sine function is periodic and its period is $2\pi$.
View the demos

1. If $P$ is in the radius cycle 1, its ordinate can only vary from $-1$ to $+1$; and each value in that range is ordered from some point in the cycle.

2 and 3. In the 1st and 2nd quadrants, $P$ is above the $u$ axis and its ordinate is positive; in the 3rd and 4th, $P$ is below $u$ and the ordinate is negative.

4 and 5. If $x$ runs through the 1st quadrant, $P$ runs through the arc $\overset{\frown}{AB}$ and its ordinate grows from 0 to 1; in the 4th quadrant, $P$ goes from $B'$ to $A$ and the ordinate increases from $-1$ to 0. If $x$ runs through the 2nd quadrant, $P$ runs through $\overset{\frown}{BA'}$ and the ordinate decreases from 1 to 0; A similar fact occurs in the 3rd quadrant.

6. If $\operatorname{sen} x = \overline{OP_1}$ and $k \in \mathbb{Z}$, then $\operatorname{sen}(x + 2k\pi) = \overline{OP_1}$, since $x$ and $x + 2k\pi$ have the same image $P$ in the cycle. Therefore, for every real $x$,

$$\operatorname{sen} x = \operatorname{sen}(x + 2k\pi),$$

and the sine function is periodic. The period is the smallest positive value of $2k\pi$, that is, $2\pi$ (no $p$ between 0 and $2\pi$ will do, as $\operatorname{sen}\frac{\pi}{2} = 1$ only repeats after a full revolution).

Graph: the sinusoid

Let $x$ traverse the interval $[0, 2\pi]$. When the image $P$ turns completely counterclockwise, the ordinate of $P$ varies like this:

$x$ $0$ $\frac{\pi}{2}$ $\pi$ $\frac{3\pi}{2}$ $2\pi$
$\operatorname{sen} x$ $0$ grows $1$ decreases $0$ decreases $-1$ grows $0$

With $x$ on the abscissa and $\operatorname{sen} x$ on the ordinate, we obtain the graph called sinusoid:

Move the control (or use ▶ animate): the height of the cycle point is transported to the graph, which is being drawn.
The sinusoid. The rectangle $[0, 2\pi] \times [-1, 1]$ contains exactly one period; its dimensions are $2\pi \times 2$, that is, about $6{,}28 \times 2$.

As the domain of the sine function is $\mathbb{R}$, the sinusoid continues to the right of $2\pi$ and to the left of $0$, repeating the same stamp.

Variations of the sinusoid

The exercises in this chapter ask for graphs of functions such as $-\operatorname{sen} x$, $2\operatorname{sen} x$, $|\operatorname{sen} x|$, $\operatorname{sen} 2x$, $\operatorname{sen}\frac{x}{2}$, $1 + \operatorname{sen} x$, and $\operatorname{sen}\left(x - \frac{\pi}{4}\right)$. They all follow the same method, in three steps:

  1. we assign convenient values ​​to the sine argument ($0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$);
  2. we calculate the sine of these values;
  3. we calculate $x$ and $y$ from them.

ResumoImage and period of f(x) = a + b·sin(cx + d)

For real $a, b, c, d$ with $b \ne 0$ and $c \ne 0$:

$$\operatorname{Im}(f) = \big[a - |b|,\ a + |b|\big] \qquad\qquad p(f) = \frac{2\pi}{|c|}.$$
  • $a$ translates the graph up ($a > 0$) or down ($a < 0$);
  • $b$ stretches the graph vertically (the amplitude is $|b|$) and, if $b < 0$, reflects it on the $x$ axis;
  • $c$ compresses the graph horizontally: the period becomes $\frac{2\pi}{|c|}$;
  • $d$ shifts the graph horizontally: a period starts at $x = -\frac{d}{c}$.

The demonstration is in exercise 43 of chapter list.

Move the controls and observe the effect of each coefficient. Use the values ​​from an exercise to check your graph.