Compare coordinates

In the first quadrant, the straight line of the radius that passes through $P=(\cos x,\sin x)$ has a slope equal to the height divided by the horizontal displacement. We extend this ratio to all angles where the denominator is not zero:

$$\tan x=\frac{\sin x}{\cos x}.$$

Logo:

$$\operatorname{Dom}(\tan)=\mathbb R\setminus \left\{\frac\pi2+k\pi:k\in\mathbb Z\right\}.$$

The image is $\mathbb R$. The tangent is positive in quadrants I and III and negative in quadrants II and IV. Its zeros are $x=k\pi$.

Statement of the period

After a half-turn, the two coordinates change sign:

$$\tan(x+\pi)=\frac{-\sin x}{-\cos x}=\tan x.$$

Thus, $\pi$ is a period. For a positive period $T$ to repeat the value of $\tan0=0$, it needs to be a positive multiple of $\pi$; the smallest is $\pi$. The tangent is also odd, because:

$$\tan(-x)=\frac{-\sin x}{\cos x}=-\tan x.$$

The graph is made of branches

In each interval $(-\pi/2+k\pi,\pi/2+k\pi)$, the tangent is strictly increasing. When the horizontal coordinate approaches zero without the vertical coordinate canceling out, the quotient increases in magnitude without limit.

The lines $x=\pi/2+k\pi$ are vertical asymptotes. They are not points on the graph. There is no “infinite tangent” as a real value in $\pi/2$; the function is simply not defined in that input.

A branch passes through $(-\pi/4,-1)$, $(0,0)$ and $(\pi/4,1)$. The rest are obtained by translations of $\pi$.

A restriction on algebraic manipulation

The equality $\tan x\cos x=\sin x$ holds where $\tan x$ exists. The simplified product does not allow using the original expression in $x=\pi/2$, even if the sine is defined there.

In everyday life

  • Slope in percentage. A street with 10% slope has $\operatorname{tg}\alpha = 0{,}10$, that is, $\alpha \approx 5{,}7^\circ$.
  • Field of view. A camera with an aperture of $90^\circ$ sees, at 3 m away, a width of $2 \cdot 3\operatorname{tg} 45^\circ = 6$ m.
  • Why the tangent “explodes”. Close to $90^\circ$ the slope becomes a wall: the ramp would have to rise infinitely for each meter walked.
The width that a camera sees at a distance $d$ is $2d\operatorname{tg}\beta$.

Exercises with resolution

1. Calculate $\tan(3\pi/4)$ and $\tan(7\pi/4)$.

Check resolution 1

In both cases the module is 1 and the coordinate signs are opposite. Both are worth $-1$.

2. What is the fundamental period of $g(x)=\tan(3x)$?

Check resolution 2

$\pi/3$. Prohibited points satisfy $3x=\pi/2+k\pi$, or $x=\pi/6+k\pi/3$.

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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