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 [at original publication], 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,

$$\sin 18° = \frac {\sqrt{5} - 1}{4}.$$