The Tangent Function
Introduced| English |
|---|
| tangent/ˈtændʒənt/ |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ |
| period/ˈpɪərɪəd/ |
A wave that breaks
- Sine and cosine roll along smoothly, forever bounded between $-1$ and $1$.
- The tangent function is wilder: it shoots up to infinity, then restarts.
- Its graph is a chain of separate branches, each broken by an invisible wall.
- Understanding why it breaks reveals how it is built from sine and cosine.
Tangent is a ratio
- The tangent 正切 is defined as $\tan x = \dfrac{\sin x}{\cos x}$.
- On the unit circle it is the slope of the line from the origin to the point.
- Where sine is $0$, tangent is $0$ (the branches cross the axis).
- Where cosine is $0$, the ratio divides by zero — and that is where it breaks.
The tangent function equals…
By definition, $\tan x = \dfrac{\sin x}{\cos x}$ — sine over cosine.
What is $\tan 0$?
$\tan 0 = \dfrac{\sin 0}{\cos 0} = \dfrac{0}{1} = 0$.
Asymptotes where cosine vanishes
- A vertical asymptote 竖直渐近线 appears wherever $\cos x = 0$.
- That happens at $x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \dots$ — every half-turn.
- Approaching an asymptote, the output races off to $+\infty$ or $-\infty$.
- The graph never crosses these walls; it just climbs steeply beside them.
Tangent as a slope that runs away
On the unit circle, tangent is the slope of the radius line. As the angle nears 90 degrees, that slope shoots toward infinity.
The tangent function has a vertical asymptote wherever…
Where $\cos x = 0$, the fraction $\tfrac{\sin x}{\cos x}$ divides by zero — a vertical asymptote.
Select all true statements about $y = \tan x$.
Unlike sine and cosine, tangent is unbounded — it shoots to $\pm\infty$. The other three are correct.
A shorter period
- Sine and cosine repeat every $2\pi$, but the period 周期 of tangent is just $\pi$.
- So tangent completes a full cycle twice as often.
- Within each period it climbs steadily from $-\infty$ up to $+\infty$.
- One branch spans from one asymptote to the next.
Unlike sine and cosine, the tangent function has period ____ (not $2\pi$).
Tangent repeats every $\pi$ — twice as often as sine or cosine.
Unbounded output
- Unlike sine and cosine, tangent has no maximum or minimum.
- Its range is all real numbers — it reaches every height.
- Near an asymptote it grows without limit; at the middle of a branch it is $0$.
- This unbounded, repeating shape models steepness and angles of elevation.
Tangent is undefined at its asymptotes — $\tan\tfrac{\pi}{2}$ has no value, because $\cos\tfrac{\pi}{2} = 0$ and you cannot divide by zero. Do not try to plug those angles in; the function simply does not exist there.
Find $\tan\tfrac{\pi}{4}$ and explain why $\tan\tfrac{\pi}{2}$ is undefined.
- $\tan\tfrac{\pi}{4} = \dfrac{\sin(\pi/4)}{\cos(\pi/4)} = \dfrac{\sqrt2/2}{\sqrt2/2} = 1$.
- $\tan\tfrac{\pi}{2} = \dfrac{\sin(\pi/2)}{\cos(\pi/2)} = \dfrac{1}{0}$, which is undefined.
- So $x = \tfrac{\pi}{2}$ is a vertical asymptote, not a point on the graph.
The tangent function is $\tan x = \tfrac{\sin x}{\cos x}$. It has a vertical asymptote wherever $\cos x = 0$, a period of $\pi$ (not $2\pi$), and an unbounded range — each branch climbs from $-\infty$ to $+\infty$ and repeats.