MATH: Understanding trigonometric functions as the root of polynomials

By Bruno

When I started learning trigonometric functions (sine, cosine, tangent ...) I always could find some angle that its sine and cosine(and tangent) were almost impossible to calculate using the sum and subtraction formulas and given only the special angles (30°, 45° and 60°). A good example of a hard one is 20°! Note that 3x20°=60°, so we would like to have a formula of tan(3x) in function of x. By now, we don't have tan(3x), but we do have tan(2x+x), and we also have tg(2x):

And so, if u= tanx and x = 20, then:

It means that tan 20° is one of the roots of the polynomial above, actually all the numbers with tan3x= tan 60° are roots of this polynomial, the others are tan 140° and tan 260°. I will let as an exercise for the reader to calculate these tangent values ;).

But isn't there a way to find a polynomial that expresses tan x/n (n a positive integer) without using these ugly formulas? The answer is yes, and the approach is by complex numbers. It is well-known that if e is the euler constant and i is the square root of -1, then

and so,

by expanding the right hand side using the newton binomial theorem:

Now we separate this sum into two:

If n is even and

If n is odd, but either way, we can make (real part of the Left hand side)=(real part of the Right hand side), and the same thing with the imaginary part, so that,

Alright! Now things are starting to look nice. However, we still not there, we want tan(nx), so remember that tan(nx)=sin(nx)/cos(nx).......hmmm... I think that dividing the first equation above by the second one is a good idea! Let's try:

With this, we got tan(nx) in function of sin x and cos x. AGAIN, remember that tan(x)=sin(x)/cos(x)... why don't we divide the numerator and the denominator by (cosx)^n?

hehe.... we almost got what we claimed! But from now on, things are really easy, just let

and

then,

where S is a polynomial where it's roots are all the numbers tan(x) such that

And we are done. ;)

Final observation: normally, you won't see this kind of thing in math olympiads (they are quite trivial...), however, this idea of turning tan(xn) into a polynomial can be useful for calculating sum of tangents and other stuff...