From movement to a function

A function associates a single output with each input of the domain. When making a point go through the trigonometric cycle, we can observe its height, its horizontal position or other geometrically constructed quantities. Each choice produces a circular function.

The input will generally be a real number representing the spin in radians. Even if two different inputs lead to the same point, each has a unique image per function. A function can repeat outputs.

What does repeat mean?

We say that $T>0$ is a period when, for all allowed inputs, the domain and values ​​repeat by this offset:

$$f(x+T)=f(x).$$

When there is a smallest positive period, it is called fundamental period. It is necessary to distinguish “a period” from “the smallest period”: if $T$ is a period, $2T$ and $3T$ are also.

A constant function admits all $T>0$ as a period and has no smallest positive period. This shows why the existence of the minor needs to be examined.

Demonstration of repetition in the cycle

The point of $x+2\pi$ coincides with that of $x$. Therefore, any coordinate of this point is repeated after $2\pi$. Sine and cosine, which will be defined in the next classes, satisfy:

$$\sin(x+2\pi)=\sin x,\qquad\cos(x+2\pi)=\cos x.$$

The graph of a periodic function can be constructed by repeating a section corresponding to a period. The ends of this section must be compatible with the definition of the function; points where it does not exist remain absent.

Change input speed

If $g(x)=\sin(bx+c)$ with $b\ne0$, we want the change of $x$ to produce a complete turn in the argument:

$$|b|T=2\pi\quad\Rightarrow\quad T=\frac{2\pi}{|b|}.$$

For $g(x)=a\sin(bx+c)+d$, with $a\ne0$, this remains the fundamental period. The amplitude is $|a|$ and the mean axis is $y=d$. If $a=0$, the function is constant and the fundamental period rule is no longer valid.

For example, $g(x)=2\sin(3x)-1$ oscillates from $-3$ to 1 and has period $2\pi/3$.

In everyday life

  • Heart. With 75 beats per minute, the cardiac cycle has a period of $\frac{60}{75} = 0{,}8$ s.
  • Outlet. Brazilian alternating current has a frequency of 60 Hz: period of $\frac{1}{60}$ s.
  • Seasons. The length of the day and the average temperature repeat with a period of about 365,25 days.
Heart, socket, traffic light, Moon and week: each phenomenon with its period.

Exercises with resolution

1. What is the fundamental period of $f(x)=\cos(4x)$?

Check resolution 1

$2\pi/4=\pi/2$.

2. Does the function $f(x)=5$ have a fundamental period?

Check resolution 2

No. Every positive number is a period, and there is no smallest positive real number.

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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