Your math teacher never taught you these because they're just labels attached to quantities expressible in terms of sine and cosine. They don't aid mathematical understanding, and they don't let you solve any problem that you couldn't before.
But they did have a purpose, years ago. Before pocket calculators, when people used slide rules and log tables, a table of haversines helped avoid nasty loss-of-precision errors near the roots of a function.
> I must admit I was a bit disappointed when I looked these up. They’re all just simple combinations of dear old sine and cosine. Why did they even get names?! ...
> The secret trig functions, like logarithms, made computations easier. Versine and haversine were used the most often. Near the angle θ=0, cos(θ) is very close to 1. If you were doing a computation that had 1-cos(θ) in it, your computation might be ruined if your cosine table didn’t have enough significant figures.
/s
... which is why I plan to ignore tangent from now on.
But notice they are at least not polynomial in cos and sin, whereas the functions introduced in this article are.
To be erudite about it, there are essentially no other trig functions than cos and sin, in the sense that cos and sin (aka x and y) generate the ring of polynomial functions on the circle, ℂ[x,y]/(x²+y²-1).
Versine etc. are elements of the ring and therefore they are generated (as polynomials) by cos and sin, whereas tan etc. are not elements of the ring because they sometimes go to infinity (they're elements of the larger function field).
It might be a little unfair to expect your trig teacher to prove that to you.