The Tangent Function · 正切函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| tangent/ˈtændʒənt/ | 正切 | zhèng qiē |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
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.
一条会断裂的波
- 正弦和余弦平滑地滚动,永远被限制在 $-1$ 和 $1$ 之间。
- 正切函数更狂野:它冲向无穷,然后重新开始。
- 它的图像是一串分开的分支,每一支都被一堵看不见的墙隔断。
- 理解它为什么断裂,就揭示了它如何由正弦和余弦搭建而成。
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.
正切是一个比值
- 正切(tangent)定义为 $\tan x = \dfrac{\sin x}{\cos x}$。
- 在单位圆上,它是从原点到该点的直线的斜率。
- 在正弦为 $0$ 处,正切为 $0$(各分支穿过轴)。
- 在余弦为 $0$ 处,这个比值除以零——那就是它断裂的地方。
The tangent function equals… · 正切函数等于……
By definition, $\tan x = \dfrac{\sin x}{\cos x}$ — sine over cosine. · 根据定义,$\tan x = \dfrac{\sin x}{\cos x}$——正弦除以余弦。
What is $\tan 0$? · $\tan 0$ 是多少?
$\tan 0 = \dfrac{\sin 0}{\cos 0} = \dfrac{0}{1} = 0$. · $\tan 0 = \dfrac{\sin 0}{\cos 0} = \dfrac{0}{1} = 0$。
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.
余弦消失处的渐近线
- 在任何 $\cos x = 0$ 的地方,都会出现一条竖直渐近线(vertical asymptote)。
- 这发生在 $x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \dots$——每隔半圈一次。
- 靠近一条渐近线时,输出飞速冲向 $+\infty$ 或 $-\infty$。
- 图像从不越过这些墙;它只是紧挨着它们陡峭地爬升。
Tangent as a slope that runs away · 作为斜率、会飞奔的正切
On the unit circle, tangent is the slope of the radius line. As the angle nears 90 degrees, that slope shoots toward infinity. · 在单位圆上,正切是半径线的斜率。当角接近 90 度时,那个斜率冲向无穷。
The tangent function has a vertical asymptote wherever… · 正切函数在……处有一条竖直渐近线。
Where · 何地 $\cos x = 0$, the fraction $\tfrac{\sin x}{\cos x}$ divides by zero — a vertical asymptote. · 在 $\cos x = 0$ 处,分数 $\tfrac{\sin x}{\cos x}$ 除以零——一条竖直渐近线。
Select all · 所有 true statements about $y = \tan x$. · 选出关于 $y = \tan x$ 的所有正确说法。
Unlike sine and cosine, tangent is unbounded — it shoots to $\pm\infty$. The other three are correct. · 与正弦和余弦不同,正切是无界的——它冲向 $\pm\infty$。其余三条正确。
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.
更短的周期
- 正弦和余弦每隔 $2\pi$ 重复,但正切的周期(period)只有 $\pi$。
- 所以正切完成一个完整循环的频率是它们的两倍。
- 在每个周期内,它从 $-\infty$ 稳步爬升到 $+\infty$。
- 一支从一条渐近线跨到下一条。
Unlike sine and cosine, the tangent function has period ____ (not $2\pi$). · 与正弦和余弦不同,正切函数的周期是____(不是 $2\pi$)。
Tangent repeats every $\pi$ — twice as often as sine or cosine. · 正切每隔 $\pi$ 重复一次——比正弦或余弦频繁一倍。
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.
无界的输出
- 与正弦和余弦不同,正切没有最大值或最小值。
- 它的值域是全体实数——它能到达任何高度。
- 靠近渐近线时它无限增长;在一支的中间它是 $0$。
- 这种无界、重复的形状用来给陡度和仰角建模。
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.
正切在它的渐近线处是无定义的——$\tan\tfrac{\pi}{2}$ 没有值,因为 $\cos\tfrac{\pi}{2} = 0$,而你不能除以零。不要试图代入那些角度;函数在那里根本不存在。
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.
求 $\tan\tfrac{\pi}{4}$,并解释为什么 $\tan\tfrac{\pi}{2}$ 无定义。
- $\tan\tfrac{\pi}{4} = \dfrac{\sin(\pi/4)}{\cos(\pi/4)} = \dfrac{\sqrt2/2}{\sqrt2/2} = 1$。
- $\tan\tfrac{\pi}{2} = \dfrac{\sin(\pi/2)}{\cos(\pi/2)} = \dfrac{1}{0}$,这是无定义的。
- 所以 $x = \tfrac{\pi}{2}$ 是一条竖直渐近线,而不是图像上的一个点。
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.
正切函数是 $\tan x = \tfrac{\sin x}{\cos x}$。在任何 $\cos x = 0$ 处它有一条竖直渐近线,周期是 $\pi$(不是 $2\pi$),值域无界——每一支从 $-\infty$ 爬到 $+\infty$ 并重复。