If the sine is the height of the rotating point, the cosine is its horizontal distance from the center. The two go together: when one is at maximum, the other goes through zero.

The story

For centuries, the cosine had no name of its own: it was just “the sine of the complement”. Viète called it sinus residuae, and each author wrote differently. In 1620, the Englishman Edmund Gunter proposed, in his Canon triangulorum, the abbreviated form co-sinus — and with it also co-tangens. Four years later, Gunter was the first to use the abbreviation sin in a figure; The names stuck, and the prefix co- became the brand of all “complement” functions.

Gunter’s chain: with it, the acre — the unit of area for English surveyors — becomes exactly 10 square chains.

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

In everyday life

  • Solar panels. The energy that a panel receives is proportional to the cosine of the angle between the Sun’s rays and the perpendicular to the panel: with the rays at $60^\circ$ from this perpendicular, it receives half of what it would receive from the front.
  • Engines. In the connecting rod and crank mechanism, the piston position approximately follows $r\cos\theta$, where $\theta$ is the crankshaft rotation.
  • Ramps. A ramp of length $L$ and inclination $\alpha$ occupies a horizontal length $L\cos\alpha$ on the floor — this is the space that the project needs to reserve.
A solar panel receives energy proportional to the cosine of the angle between the rays and the perpendicular to the panel.

Theory

Definition

DefiniçãoCosine and cosine function

Given a real number $x$, let $P$ be its image in the cycle. The cosine of $x$ (we indicate $\cos x$) is the abscissa $\overline{OP_2}$ of the point $P$ in relation to the system $uOv$.

The cosine function is the function $f\colon \mathbb{R} \to \mathbb{R}$ that associates each real $x$ with the real $\overline{OP_2} = \cos x$, that is, $f(x) = \cos x$.

Drag $P$: the segment $\overline{OP_2}$, on the cosine axis, is $\cos x$. Change to sine and compare.

Properties

PropriedadeProperties of the cosine function

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

These are the same ideas used for the sine, now looking at the abscissa of $P$.

1. The abscissa of a point of the radius cycle 1 ranges from $-1$ to $1$.

2 and 3. In the 1st and 4th quadrants, $P$ is to the right of the $v$ axis and has a positive abscissa; in the 2nd and 3rd, it is on the left and has a negative abscissa.

4 and 5. When $x$ passes through the 1st quadrant, $P$ goes from $A$ to $B$ and the abscissa decreases from 1 to 0; in the 2nd, from $B$ to $A'$, it decreases from 0 to $-1$. In the 3rd, $P$ goes from $A'$ to $B'$ and the abscissa grows from $-1$ to 0; in the 4th, from $B'$ to $A$, it grows from 0 to 1.

6. $x$ and $x + 2k\pi$ have the same image $P$, so $\cos x = \cos(x + 2k\pi)$ for all $x$ and all $k \in \mathbb{Z}$. The smallest positive value of $2k\pi$ is $2\pi$.

Graph: the cosine

When $x$ goes through $[0, 2\pi]$ and $P$ makes a complete turn counterclockwise, the abscissa of $P$ varies like this:

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

The graph of $f(x) = \cos x$ is called cosineoid:

The cosine. The highlighted rectangle, with dimensions $2\pi \times 2$, contains a period.
The cosine is the abscissa of the cycle point; the graph records this value for each $x$. Compare with the sinusoid: the cosineoid is the sinusoid shifted from $\frac{\pi}{2}$ to the left.

Since the domain is $\mathbb{R}$, the cosine continues to the right of $2\pi$ and to the left of $0$.