What is the purpose, if any, of secant, cosecant, and cotangent? I’m an engineering student so I use sine, cosine, and tangent a lot, but almost never the other 3. Creating a new trig function just to be the reciprocal seems to be a bit redundant.
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In engineering, these functions are used to solve design and construction problems. For example, in the construction of bridges and tall buildings, these functions are used to calculate the lengths of cables required to support the weight of the structure. They are also used in electrical engineering to calculate the impedance of electrical circuits.
That you have been taught these named functions is due to historical inertia. In some countries these names are not used anymore. Henri Cohen wrote the following in the preface of his book “Number Theory Volume II: Analytic and Modern Tools”:
> The trigonometric functions sec x and csc x do not exist in France, so I will not use them.
People in France who need sec(x) have no problem: they will just write 1/cos(x). And they may learn derivative formulas like (tan x)’ = 1/(cos x)^2 instead of (tan x)’ = sec^(2)(x). [Note: If anyone who learned high school math in France is reading this, is Henri Cohen’s remark accurate: you don’t get taught sec x and csc x in school?]
Back in the 19th century there were named variants on trigonometric functions that are no longer in use, such as versine, coversine, and haversine: https://en.wikipedia.org/wiki/Versine. The function versin(x) is 1 – cos(x), which looks about as stupid as having a special name for 1/cos(x). The reason there was ever a special name for 1 – cos(x) is that its values were important enough to be worth talking about and tabulating, e.g., in *navigation*. (Many functions that show up repeatedly get named: the Error function, the Gamma function, and so on.)
See the History section of the page https://en.wikipedia.org/wiki/Integral_of_the_secant_function for a link between integrating the function sec(x) and nautical tables. While navigation has not disappeared, it is not done with printed look-up tables anymore and we’re no longer taught function names like versine. Maybe in 100 years the function name sec(x) will be dropped worldwide, but don’t count on it.
I assure you that your math instructors are as uninterested in teaching those extra trig functions as you probably are in learning them, at least for cot(x) and csc(x). When I teach calculus, I only mention cot(x) and csc(x) in passing because they occur in the book, but I tell the students that they will not appear on any homework or exam problems. We use sec(x) in the course due to its appearance in the derivative of tan(x) (and also (sec x)’ = (sec x)(tan x)). Have you never had to differentiate tan(x) when solving some ODE?
By the way, you’re forgetting about many other modern trigonometric functions: https://www.theonion.com/nation-s-math-teachers-introduce-27-new-trig-functions-1819575558
As far as I know, its due to how often they appear across maths – it helps to define them as their own thing(s) due to how frequently you have to write or use them. For example, they appear in the trig identities tan^2 x + 1 = cosec^2 x & 1 + cot^2 x = sec^2 x, and also often in calculus, like for example when considering d/dx(tanx) = sec^2 x. You could make the argument that anything in maths can be written as its underlying mathematical representation rather than being a separate entity with its own name, but as soon as it becomes frequently used, it simply becomes a matter of convenience to name it. Sec, csc, and cotan are no different.
I actually used the cotangent function in a scientific article because it was more convenient, in writing the equation, to show multiplication by the cotangent than division by the tangent. In that instance I was calculating the area of a right triangle based on the trapezoid created by truncating the larger triangle.
You don’t *have* to use them explicitly. The functions sec, cosec and cot are just 1/cos, 1/sin and 1/tan respectively, and indeed some prefer to just use the latter. But they’re traditional names for completeness’s sake and the old trig and differentiation/integration formulas are usually still written with them, so people still use them.
It’s like ‘What’s the point of this obscure word that has a synonym, this random irregular verb or quirk of English spelling?’ You could come up with a more streamlined version of the language without, but you’ll need to understand them when others use them.
These terms were first used in geometric contexts before the concept of a function emerged. Instead of thinking of sine et al. as functions, everything in trigonometry would be described in terms of geometric figures in which specific lines were labelled as the sine, secant, etc. Once people started thinking in terms of ratios and then functions instead of lengths of lines, the terminology carried over, even if some of it now seems slightly pointless. But sec, csc and cot aren’t that hard to remember, and are slightly easier to write and say than 1/sin, etc., so there isn’t much incentive to get rid of them.
Tbh it’s fairly common for mathematicians to introduce new notation for slight variants of functions. For example, you often see sinc(x) as a shorthand for sin(x)/x, and you have stuff like the digamma function, which is just the derivative of the logarithm of the gamma function. Sometimes you get complicated expressions with many copies of a given function, so these little simplifications add up, and it can be easier to talk in general terms about “the sinc function” than “sin(x)/x for some x”. But I don’t know if anyone would have bothered to introduce notation for the reciprocal trig functions if it didn’t already exist for historical reasons.
Well, the ratio of the hypotenuse to the adjacent side is always going to be the same for a particular angle. So you gotta name it something. And Inv Cos means something else.
In the days before calculators, they had massive long lookup tables for these kinds of functions. Having extras for the reciprocals made sense, because finding the reciprocal of a rounded decimal is tedious and inaccurate, especially with the risk of human error in there.
These days, they’re basically historical oddities that are rarely used except when they make notation simpler (which I presume is also the reason they got individual names in the first place).
Basically, it wasn’t redundant when the alternative was long division by hand.
The most basic place where sec(x) occurs in math if you at first only care about sin(x), cos(x), and tan(x) is in the derivative formula
(tan x)’ = sec^(2)(x).
Trigonometric functions have hyperbolic trig analogues (essentially replace x with ix) and the hyperbolic trig function
sech(pi x) = 2/(e^(pi x) + e^(-pi x)) is interesting in Fourier analysis since it is equal to its Fourier transform (using a suitable normalization of that transform).
In higher math, I can think of two places where the function cot(x) has a purpose:
1. The Riemann zeta function can be calculated at positive even numbers by computing a series expansion for cot(x) in two ways.
2. The q-expansion of Eisenstein series for the group SL(2,Z) is derived using the function pi cot(pi x).
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