Secant, Cosecant, and Cotangent · 正割、余割与余切
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| reciprocal/rɪˈsɪprəkl/ | 倒数 | dào shǔ |
| secant/ˈsiːkənt/ | 正割 | zhèng gē |
| cosecant/ˈkəʊsekənt/ | 余割 | yú gē |
| cotangent/ˈkəʊtændʒənt/ | 余切 | yú qiē |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
Three more from flipping
- Sine, cosine, and tangent have three partners built by flipping them over.
- Each new function is simply $1$ divided by one of the originals.
- They appear whenever a ratio is easier written "upside down".
- Their graphs inherit their asymptotes from wherever the originals hit zero.
翻转出来的另外三个
- 正弦、余弦和正切各有一个搭档,由把它们翻过来构成。
- 每个新函数不过是 $1$ 除以其中一个原函数。
- 每当一个比值写成"上下颠倒"更方便时,它们就登场。
- 它们的图像从原函数为零的地方继承了渐近线。
The three reciprocals
- Secant 正割 is the reciprocal 倒数 of cosine: $\sec x = \dfrac{1}{\cos x}$.
- Cosecant 余割 is the reciprocal of sine: $\csc x = \dfrac{1}{\sin x}$.
- Cotangent 余切 is the reciprocal of tangent: $\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}$.
- A handy memory hook: "co-" and the base function never match up (secant pairs with cosine).
三个倒数
- 正割(secant)是余弦的倒数(reciprocal):$\sec x = \dfrac{1}{\cos x}$。
- 余割(cosecant)是正弦的倒数:$\csc x = \dfrac{1}{\sin x}$。
- 余切(cotangent)是正切的倒数:$\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}$。
- 一个好用的记忆提示:带"余"的和同名的基础函数从不配对(正割配余弦)。
The cosine wave whose reciprocal is secant · 其倒数是正割的余弦波
y = cos x
Secant is 1 divided by cosine. Wherever this cosine wave crosses zero, secant blows up to an asymptote. · 正割是 1 除以余弦。这条余弦波在哪里穿过零,正割就在哪里爆炸成一条渐近线。
The secant function is defined as… · 正割函数定义为……
Secant is the reciprocal · 倒数 of cosine: $\sec x = \dfrac{1}{\cos x}$. · 正割是余弦的倒数:$\sec x = \dfrac{1}{\cos x}$。
Which pairing is correct? · 下面哪个配对是正确的?
Cosecant is $1/\sin$ and cotangent is $1/\tan = \cos/\sin$ — the reciprocals of sine and tangent. · 余割是 $1/\sin$,余切是 $1/\tan = \cos/\sin$——正弦和正切的倒数。
Select all · 所有 correct reciprocal identities. · 选出所有正确的倒数关系。
Secant is $1/\cos$, not $\sin$. The other three are correct reciprocal definitions. · 正割是 $1/\cos$,不是 $\sin$。其余三条是正确的倒数定义。
Where they blow up
- A reciprocal is undefined when the original is zero — you cannot divide by zero.
- So secant has a vertical asymptote 竖直渐近线 wherever $\cos x = 0$.
- Cosecant blows up wherever $\sin x = 0$; cotangent wherever $\sin x = 0$ too.
- Between the asymptotes, each curve swoops up toward $\pm\infty$.
它们在哪里爆炸
- 倒数在原函数为零时无定义——你不能除以零。
- 所以正割在任何 $\cos x = 0$ 处有一条竖直渐近线(vertical asymptote)。
- 余割在任何 $\sin x = 0$ 处爆炸;余切也在 $\sin x = 0$ 处爆炸。
- 在渐近线之间,每条曲线都俯冲向 $\pm\infty$。
Secant has a vertical ____ wherever $\cos x = 0$, because you cannot divide by zero. · 正割在任何 $\cos x = 0$ 处都有一条竖直____,因为你不能除以零。
Since $\sec x = 1/\cos x$, a zero cosine makes secant undefined — a vertical asymptote. · 因为 $\sec x = 1/\cos x$,余弦为零使正割无定义——一条竖直渐近线。
Secant's output never lies strictly between $-1$ and $1$. · 正割的输出从不严格地介于 $-1$ 和 $1$ 之间。
Since $|\cos x| \le 1$, its reciprocal $|\sec x| \ge 1$ — secant is always at least $1$ in size. · 因为 $|\cos x| \le 1$,它的倒数 $|\sec x| \ge 1$——正割的大小总是至少为 $1$。
Their shape
- Because $|\cos x| \le 1$, its reciprocal secant satisfies $|\sec x| \ge 1$.
- So secant never enters the strip between $-1$ and $1$ — it lives outside it.
- Each secant "U" sits above a cosine peak or below a cosine trough.
- Cosecant looks the same, shifted to match sine instead of cosine.
它们的形状
- 因为 $|\cos x| \le 1$,它的倒数正割满足 $|\sec x| \ge 1$。
- 所以正割从不进入 $-1$ 和 $1$ 之间那条带子——它住在外面。
- 每一个正割的"U"形都位于余弦波峰之上或波谷之下。
- 余割看起来相同,只是平移到与正弦而非余弦匹配。
Why they are useful
- Some formulas are cleaner with secant or cotangent than with a fraction of sine and cosine.
- The Pythagorean identities come in secant and cosecant forms too.
- In calculus, the derivative of tangent is $\sec^2 x$ — the reciprocals show up naturally.
- Knowing all six trig functions lets you pick the simplest one for the job.
它们为什么有用
- 有些公式用正割或余切,比用正弦余弦的分数更简洁。
- 勾股恒等式也有正割和余割的形式。
- 在微积分里,正切的导数是 $\sec^2 x$——这些倒数自然地出现。
- 掌握全部六个三角函数,能让你为手头的任务挑出最简单的那个。
Do not confuse the reciprocal $\sec x = \tfrac{1}{\cos x}$ with the inverse $\cos^{-1} x = \arccos x$. Despite the similar names, one is "one over cosine" and the other is "the angle with this cosine" — completely different functions.
不要把倒数 $\sec x = \tfrac{1}{\cos x}$ 与反函数 $\cos^{-1} x = \arccos x$ 搞混。尽管名字相似,一个是"一除以余弦",另一个是"余弦等于此值的那个角"——完全不同的函数。
Evaluate $\sec\tfrac{\pi}{3}$ and explain why $\csc 0$ is undefined.
- $\cos\tfrac{\pi}{3} = \tfrac12$, so $\sec\tfrac{\pi}{3} = \tfrac{1}{1/2} = 2$.
- $\csc 0 = \tfrac{1}{\sin 0} = \tfrac{1}{0}$, which is undefined.
- So $x = 0$ is a vertical asymptote of cosecant.
求 $\sec\tfrac{\pi}{3}$,并解释为什么 $\csc 0$ 无定义。
- $\cos\tfrac{\pi}{3} = \tfrac12$,所以 $\sec\tfrac{\pi}{3} = \tfrac{1}{1/2} = 2$。
- $\csc 0 = \tfrac{1}{\sin 0} = \tfrac{1}{0}$,这是无定义的。
- 所以 $x = 0$ 是余割的一条竖直渐近线。
Secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent. Each has a vertical asymptote wherever its base function is zero, and secant/cosecant outputs never fall strictly between $-1$ and $1$.
正割、余割和余切是余弦、正弦和正切的倒数。每一个在其基础函数为零处都有一条竖直渐近线,而正割/余割的输出从不严格落在 $-1$ 和 $1$ 之间。