Saturday, December 13, 2014

What is the Sine of 18°?

To find the exact values of sine (and cosine) of a given angle, you need to use identities. This particular value is interesting in that it takes a fair bit of work to arrive at this value analytically, and yet the value is expressible as a fairly simple radical. The first thing to notice is that 90 = 5(18). You probably know the half angle identity, but not a fifth angle identity.  And for good reason—the fifth angle identity has multiple solutions; it's a fifth degree polynomial.

I did this calculation manually the first time through—some years ago. It goes a lot easier with a computer algebra system. Today, I'm going to use Maxima to solve the problem.

An identity which comes up in the development of the complex numbers, gives a convenient way of expressing $$\sin {n\theta}$$ in terms of combinations of $$\sin \theta$$ and $$\cos \theta$$.

De Moivre's formula:
$r^n (\cos \theta + i \sin \theta)^n = r^n (\cos {n\theta} + i \sin {n\theta}).$
So, if we take $$r=1$$ and $$n=5$$, we can do a binomial expansion and easily isolate the $$sin \theta$$ and $$\cos \theta$$. In a Maxima cell, I type

eq: (cos(%theta) + %i*sin(%theta))^5 = cos(5*%theta) + %i*sin(5*%theta);
realpart(eq);
imagpart(eq);

After executing, I obtain the fifth angle identities (if you want to call them that).
$\mathrm{cos}\left( 5\,\theta\right) =5\,\mathrm{cos}\left( \theta\right) \,{\mathrm{sin}\left( \theta\right) }^{4}-10\,{\mathrm{cos}\left( \theta\right) }^{3}\,{\mathrm{sin}\left( \theta\right) }^{2}+{\mathrm{cos}\left( \theta\right) }^{5}$
$\mathrm{sin}\left( 5\,\theta\right) ={\mathrm{sin}\left( \theta\right) }^{5}-10\,{\mathrm{cos}\left( \theta\right) }^{2}\,{\mathrm{sin}\left( \theta\right) }^{3}+5\,{\mathrm{cos}\left( \theta\right) }^{4}\,\mathrm{sin}\left( \theta\right)$
To eliminate $$\cos^2 \theta$$ from the sine identity we use

subst([cos(%theta)=sqrt(1-sin(%theta)^2)],%);
expand(%);

and obtain
$\mathrm{sin}\left( 5\,\theta\right) =16\,{\mathrm{sin}\left( \theta\right) }^{5}-20\,{\mathrm{sin}\left( \theta\right) }^{3}+5\,\mathrm{sin}\left( \theta\right).$
Taking $$\theta=18°$$, using $$x$$ to stand in for $$\sin 18°$$, and rearranging, we have
$0 = 16\,{x}^{5}-20\,{x}^{3}+5\,x - 1.$
Factoring this one manually was made easier by recognizing $$x = 1$$ as a root. After that I tried forming a square free polynomial and found that there was in fact a squared polynomial involved. In general, this helps in a couple ways (potentially):
1. Reduces the degree of the polynomial to be factored.
2. Sometimes produces fully reduced factors as a by-product of forming the square free polynomial.
1. This happens when each repeated factor is of different degree; that is, occurs a different number of times.
(Square free means no factors left that are raised to some power—each factor occurs exactly once.) Doing this manually involves taking the greatest common divisor between a polynomial and it's derivative. But today, let's let Maxima do it.

factor(0=16*x^5-20*x^3+5*x - 1);
$0=\left( x-1\right) \,{\left( 4\,{x}^{2}+2\,x-1\right) }^{2}.$
At this point, you really do have to be exceedingly lazy to not continue with the rest of the work manually. It's just a quadratic formula to be applied now (we discard $$x=1$$ as an artifact). To use a computer algebra system for the rest is...well...okay, it is Saturday, so...

solve(0=4*x^2+2*x-1,[x]);
$[x=-\frac{\sqrt{5}+1}{4},x=\frac{\sqrt{5}-1}{4}]$
We reject the negative solution (which is $$\sin {-54°}$$ and, interestingly, is the negative of half the Fibonacci number) on the grounds that the range of the sine function on our interval is positive. Hence,
$\sin 18° = \frac {\sqrt{5} - 1}{4}.$