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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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.
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