Is there any theory of trigonometry beyond sinusoidal curves?

Answer 1

Of course there is! In mathematics, as soon as we invent something, we start to play around with it, find variations and more conclusions, and invent more things.

After graphing sinusoidal curves y = sin x and y = cos x, then we work on the other trig functions y = tan x, y = sec x, y = csc x, and y = cot x. These are all periodic but they also have regular discontinuities (breaks) and vertical asymptotes.

We work on all the transformations of all of these graphs, y = a*f(b(x-h)) + k

We also look at combinations like y = cos ax + sin bx, y = sec x tan x , and y = sn^2 x and more

Then we learn about the *inverse functions* y = sin^(-1) x or y = arcsin x [same thing, alternate notation] and of course inverse cosine, inverse tangent, inves=rse secant, and onwards.

Somewhere along here we meet calculus and we do the derivative (slope) and integral (area under curve) of each of the above functions. We learn some peculiar and fascinating relationships between integrals of polynomial functions and inverse trig functions.

In physics we also look at combinations like y = e^-x cos x which describes a wave that "decays, anything from a bouncing bal to a spring to ripples in water.

Many people learn to work with Fourier Series (not my field but interesting) and approximate functions by an infinite sum a_0 + a_1 sin bx + a_2 sin^2 bx + a_3 sin^3 bx + . . . .

Engineers and electronics researchers often work with "square waves" which are exactly what they sound like.

When you meet imaginary and complex numbers you learn to use i = sqrt(-1) [Yes, your high school teacher said it does not exist; it is not real but imaginary.] Then you meet the absolutely fascinating equation e^(it) = cos t + i sin t, which leads to De Moivre's Theorem

e^(itn) = (cos t + i sin t)^n = cos nt + i sin nt

and the famous equation e^(i pi) + 1 = 0

There's also the field of "hyperbolic trig" functions,

sinh(x) = (e^x - e^-x)/2

cosh(x) = (e^x + e^-x)/2

Hyperbolic tangent, secant, etc can be defined from these.

They are not exactly trig functions, but they behave in oddly similar ways.

If you're asking because you're fed up with trigonometry and just want to see the end of it, ahhhhh . . . . sorry, no. If you go on with math, it will keep coming back at you, so master it now and learn to deal with it. It gets better and much more interesting.

If you're asking because you've found something you like and want to explore more, welcome!