Secant, Cosecant, and Cotangent
| English | Chinese | Pinyin |
|---|---|---|
| reciprocal | 倒数 | dào shǔ |
| secant | 正割 | zhèng gē |
| cosecant | 余割 | yú gē |
| cotangent | 余切 | yú qiē |
| vertical asymptote | 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.
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).
The cosine wave whose reciprocal is secant
y = cos x
Secant is 1 divided by cosine. Wherever this cosine wave crosses zero, secant blows up to an asymptote.
The secant function is defined as…
Secant is the reciprocal of cosine: $\sec x = \dfrac{1}{\cos x}$.
Which pairing is correct?
Cosecant is $1/\sin$ and cotangent is $1/\tan = \cos/\sin$ — the reciprocals of sine and tangent.
Select all correct reciprocal identities.
Secant is $1/\cos$, not $\sin$. The other three are correct reciprocal definitions.
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 $\pm\infty$.
Secant has a vertical ____ wherever $\cos x = 0$, because you cannot divide by zero.
Since $\sec x = 1/\cos x$, a zero cosine makes secant undefined — a vertical asymptote.
Secant's output never lies strictly between $-1$ and $1$.
Since $|\cos x| \le 1$, its reciprocal $|\sec x| \ge 1$ — secant is always at least $1$ in size.
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.
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.
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.
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.
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 $1$.