Secant, Cosecant, and Cotangent
Introduced| English | Meaning |
|---|---|
| reciprocal/rɪˈsɪprəkl/ | The value obtained by dividing one by a non-zero number. |
| secant/ˈsiːkənt/ | The reciprocal of cosine, where cosine is non-zero. |
| cosecant/ˈkəʊsekənt/ | The reciprocal of the sine of an angle, defined when that sine is nonzero. |
| cotangent/ˈkəʊtændʒənt/ | The reciprocals of cosine, sine, and tangent. |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | A vertical line approached by a curve as its output grows without bound near a particular input. |
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…
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Secant is the reciprocal of cosine: $\sec x = \dfrac{1}{\cos x}$.
Which pairing is correct?
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Cosecant is $1/\sin$ and cotangent is $1/\tan = \cos/\sin$ — the reciprocals of sine and tangent.
Select all correct reciprocal identities.
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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.
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Since $\sec x = 1/\cos x$, a zero cosine makes secant undefined — a vertical asymptote.
Secant can take any value between $-1$ and $1$.
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False. Since $|\cos x| \le 1$, its reciprocal satisfies $|\sec x| \ge 1$, so secant never lies strictly between $-1$ and $1$.
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.
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 $1$.
Write from a secant of 1
- Secant is 1 over cosine.
- Evaluate secant of 0.
What is secant of 0?
1 divided by 1 is 1.
What is secant of 0?
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The reciprocal of cosine of 0 is 1.