The height of the point

If the image of the real $x$ in the cycle is $P=(u,v)$, we define $\sin x=v$. The sine is the vertical, signed coordinate, not the unsigned distance to the horizontal axis.

Coordenadas de um ponto no ciclo trigonométrico.

Any real has an image in the cycle. The height varies from $-1$ to 1, including the extremes. Therefore:

$$\operatorname{Dom}(\sin)=\mathbb R,\qquad\operatorname{Im}(\sin)=[-1,1].$$

In each turn, the sine is positive in the upper semicircle and negative in the lower one. Its zeros are $x=k\pi$, to $k\in\mathbb Z$.

A turn of the graph

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

The sine increases from $-1$ to 1 in each interval $[-\pi/2+2k\pi,\pi/2+2k\pi]$ and decreases from 1 to $-1$ in $[\pi/2+2k\pi,3\pi/2+2k\pi]$. These sections, repeated, form the sinusoid.

Demonstration of two properties

Reflection on the horizontal axis transforms the image of $x$ into that of $-x$, maintaining the horizontal coordinate and inverting the vertical. Like this:

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

Therefore, the sine is an odd function. A complete turn preserves the point, therefore $\sin(x+2\pi)=\sin x$. The smallest positive period is $2\pi$, as the maximum height 1 only reappears after a complete revolution.

Maximums occur at $x=\pi/2+2k\pi$; the minimum ones, in $x=3\pi/2+2k\pi$.

Read a transformation

At $h(x)=4+3\sin x$, the midline is 4 and the amplitude is 3. The image is $[1,7]$. The period remains $2\pi$, as adding a constant or multiplying the output does not change the time required to complete the pattern.

In everyday life

  • Music. The A note on the tuning fork is a sinusoid of 440 cycles per second; Higher notes have a shorter period.
  • Sea waves. The profile of a regular wave, far from the beach, approaches a sinusoid; the amplitude is half the distance between the crest and the valley.
  • Vibration of the cell phone. The little motor that makes the cell phone vibrate produces a back and forth movement described by a sine.
Sounds are sinusoidal: two nearby notes added together produce a volume that pulsates.

Exercises with resolution

1. Calculate $\sin(7\pi/2)$.

Check resolution 1

Remove one turn: $7\pi/2-2\pi=3\pi/2$. The sine is $-1$.

2. What is the image of $f(x)=2-5\sin x$?

Check resolution 2

Like $-1\le\sin x\le1$, we have $-5\le-5\sin x\le5$ and $-3\le f(x)\le7$. The image is $[-3,7]$.

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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