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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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
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