The width with sign

For $P=(u,v)$ in the cycle, we define $\cos x=u$. The cosine is positive in the right semicircle and negative in the left. Its domain is $\mathbb R$ and its image is $[-1,1]$.

The axis points provide:

$x$ $0$ $\pi/2$ $\pi$ $3\pi/2$ $2\pi$
$\cos x$ $1$ $0$ $-1$ $0$ $1$

Zeros are $x=\pi/2+k\pi$. The maxima occur at $x=2k\pi$ and the minima at $x=\pi+2k\pi$, with $k\in\mathbb Z$.

Demonstration of parity and period

The points of $x$ and $-x$ are reflections on the horizontal axis. Its horizontal coordinate is equal:

$$\cos(-x)=\cos x.$$

Therefore, the cosine is even. A complete turn conserves the coordinate, so $\cos(x+2\pi)=\cos x$. For there to be a positive period $T$, in particular, $\cos T=\cos0=1$ must be valid. The smallest $T>0$ with this value is $2\pi$, which is its fundamental period.

Growth and decrease

As the point travels the upper semicircle from right to left, its horizontal coordinate falls from 1 to $-1$. Therefore, the cosine decreases in $[2k\pi,\pi+2k\pi]$. In the lower semicircle it grows, at $[\pi+2k\pi,2\pi+2k\pi]$.

Relationship with sine

A rotation of $\pi/2$ exchanges the horizontal position with the height, respecting the direction of rotation. As a consequence:

$$\cos x=\sin\left(x+\frac\pi2\right).$$

The two graphs have the same shape and are offset horizontally. It should not be concluded that their values ​​are the same for the same input: in $x=0$, the sine is 0 and the cosine is 1.

For example, $f(x)=3\cos(2x)+1$ has amplitude 3, period $\pi$, and image $[-2,4]$.

In everyday life

  • Solar panels. The power received is proportional to the cosine of the angle of incidence: it is maximum with the Sun in front and drops to zero with the Sun close to the panel.
  • Engine piston. The position of the piston follows approximately $r\cos\theta$, where $\theta$ is the rotation of the shaft.
  • Projections. The shadow of a stick of length $L$ inclined at $\alpha$, with the Sun at its head, measures $L\cos\alpha$.
The energy from a solar panel follows the cosine of the angle of incidence.

Exercises with resolution

1. Calculate $\cos(-5\pi/3)$.

Check resolution 1

By parity, it is $\cos(5\pi/3)=1/2$.

2. In which reals is the cosine worth $-1$?

Check resolution 2

$x=\pi+2k\pi$, with $k\in\mathbb Z$.

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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