Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

Question

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$.

I know that these numbers add up, but is there any specific reason for this occurrence?

Note: I discovered this in a Physics lesson, when we were studying refractive indexes, which are calculated using the formula $\frac{\sin i}{\sin r}$.

It was a coincidence that, at the time, I was discussing the Fibonacci sequence with my friend, and showing her that if you take two numbers, and calculate the next number in the series by adding the previous two numbers, like in the Fibonacci sequence, as the numbers tend toward infinity, the ratio between any two consecutive numbers in the sequence is $\phi$.

Answer 1

Here it is a nice geometric proof. Let $ABCDE$ a regular pentagon, and $F$ a point on $AC$ such that $AF=AB$. By angle chasing, we have that $CFB$ is similar to $ABC$, hence: $$ \frac{AC}{AB} = 1+\frac{CF}{AB} = 1+\frac{CF}{CB} = 1+\frac{AB}{AC}, $$ giving $\frac{AC}{AB}=\phi$. By applying the sine theorem to the triangle $ABC$ and the sine duplication formula we easily prove our claim.

Answer 2

As $\sin30^\circ=1/2$

If $y=\dfrac{\sin30^\circ}{\sin18^\circ}=\dfrac1{2\sin18^\circ},$

$$\frac1{4\sin^218^\circ}-\frac1{2\sin18^\circ}-1=\frac{1-2\sin18^\circ-4\sin^218^\circ}{4\sin^218^\circ}$$

$$=\frac{1-2\cos(90^\circ-18^\circ)-2(1-\cos36^\circ)}{4\sin^218^\circ}$$

$$=\frac{2(\cos36^\circ-\cos72^\circ)-1}{4\sin^218^\circ}=0$$

using Proving trigonometric equation $\cos(36^\circ) - \cos(72^\circ) = 1/2$

$$\implies y^2-y-1=0$$

Now clearly $y>0$