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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 $\phi$.

Taylor
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  • I don't understand the question. Do you want a proof that $\frac{\sin (30)}{\sin (18)}=\phi$ or is it something? If something else, what? – Git Gud Jan 26 '15 at 18:05
  • I am sorry if my question is unclear, but I was wondering if there was any reason to this, except that they add up.

    It is a strange question, and I am sorry if I have misled you.

     – Taylor Jan 26 '15 at 18:06
  • Use a specific formula for the area of a triangle.$E=1/2\cdot a\cdot b \cdot sin C$. Compare two areas together. – Konstantinos Gaitanas Jan 26 '15 at 18:07
  • Is this formula already known, and if not, does it have any useful implications? @GitGud – Taylor Jan 26 '15 at 18:08
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    I still have no idea what you want. To me a 'reason' is a proof. But the whole demeanor of the question suggests you're more interested not in a proof but in the intuition behind this equality. I am left then with two different interpretations of the question. – Git Gud Jan 26 '15 at 18:10
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    I believe that the latter is what I am asking for. @GitGud – Taylor Jan 26 '15 at 18:10
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    @Taylor Given that $\sin(30^\circ)$ is just $\frac12$, there are simpler ways of writing this. But it does have implications; searching on 'golden ratio pentagon' (as $\sin(18^\circ)=-\cos(108^\circ)$ and $108^\circ$ is the interior angle of a pentagon) should turn up some interesting information... – Steven Stadnicki Jan 26 '15 at 18:12
  • Oh, that is very interesting! Thank you very much! If you write this up as an answer, I will up-vote it. @StevenStadnicki – Taylor Jan 26 '15 at 18:13

2 Answers2

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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.

Jack D'Aurizio
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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 $y>0$

Community
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lab bhattacharjee
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