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The textbook we use says:

The cosecant function and the secant function are the reciprocal functions of the sine function and the cosine function, respectively, and thus are also periodic functions. Their graphs are not as useful and are seldom encountered because the cosecant is undefined at values of theta where the sine has a value of zero and the secant is undefined where the cosine has a value of zero.

I don't know of any applications for these functions outside pure mathematics. Where are they applied outside pure mathematics (if they are applied)?

colboynik
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  • They may be applied each time a reciprocal trig function appear somewhere. Secant and cosecant are not much used nowadays, but cotangent still appears quite often (it can arise when computing integrals for example). – Jean-Claude Arbaut Mar 22 '13 at 19:47
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    I don't know why we have a special name for $1/\cos{\theta}$ when we have $\cos{\theta}$ already, but personally I think it's kind of stupid. These arise everywhere trig functions occur (which is pretty much everywhere). – Alexander Gruber Mar 22 '13 at 22:13
  • I'm not sure what you mean by 'outside of pure mathematics' - the reciprocal trig functions are very useful for calculating integrals, which are pure mathematics. However, these integrals can be applied in mechanics and otherwise - would this count? – preferred_anon Mar 22 '13 at 22:42
  • By outside pure math I mean where is it applied? Mechanics would count. Where in mechanics though? – colboynik Mar 27 '13 at 19:47

2 Answers2

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Their graphs are often useful... though maybe not in rectangular coordinates. Plot $r=\sec(\theta) $ or $ r=\csc(\theta)$ in polar coordinates - you get a straight line a distance $1$ from the origin. Consequently, they are useful when you want to solve "straight line" problems in polar coordinates.

In general, polar equations are nice in physics problems because macroscopic force laws often are dependent only on the distance between two particles. You might end up using secant if you are calculating electrical interactions between a point charge (the origin) and an infinite straight charged wire at a distance $d$, parametrized by $r=d\sec(\theta)$. The contribution of any point on the wire to the electric field will depend on its distance $d\sec(\theta)$ from the origin.

Another example: if you are standing a distance $d$ from a long straight road watching a car go along, and you want to find the rate $d\theta/dt$ you have to turn your head to watch the car at a certain point, I believe you will find yourself differentiating $\sec$ or $\csc$ if you solve it as a related rates problem.

Paul Orland
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Using sec and csc is an anglo saxon tradition: In France for instance, we use ONLY cos and sin (cot occasionaly appears). I got an engineering degree in France without seeing sec and csc ever but they are averywhere in American and Australian textbooks.

I wonder what they do in other countries?

Laure

lhelmeg
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