<s>[INST] Given $\sin{x} = \frac{1 + \sqrt{5}}{4}$, find $x$ If I have been given that
$$\sin{x} = \frac{1 + \sqrt{5}}{4}$$ is there any way using which I can find out what the value of $x$ is? Sure I can remember the values, but I  was curious whether if I knew the values of $\sin{\frac{\pi}{2}}, \sin{\frac{\pi}{4}}, \sin{\frac{\pi}{3}}$ I could somehow figure out what $x$ would be? Is there any intuition behind it?
 [/INST] Let $\cos y = \frac{\sqrt5+1}4$. Then
$$\cos 2y =2\cos^2 y-1= 2\left(\frac{\sqrt5+1}4\right)^2-1=\frac{\sqrt5-1}4$$
$$\cos 4y =2\cos^2 2y-1= -\frac{\sqrt5+1}4= -\cos y= \cos(\pi\pm y)$$
The $+$ sign leads to $y=\frac\pi3$ while the $-$ sign leads to $y=\frac\pi5$. But then the value above for $\cos 2y$ is positive so we must have $y<\frac\pi4$.  Therefore, $y=\frac\pi5$ is selected and $x=\frac\pi2-\frac\pi5=\frac{3\pi}{10}$.
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