If you put $0.5^{0.5}$ into a calculator you will see that: $$0.5^{0.5}\approx0.707106781187$$ And, if you also put $\sin(\frac{\pi}{4})$ into that same calculator you will get: $$\sin(\frac{\pi}{4})\approx0.707106781187$$ Is there any specific reason why $0.5^{0.5}=\sin(\frac{\pi}{4})$, or is it just a coincidence?
$\endgroup$ 42 Answers
$\begingroup$$$0.5^{0.5}=\sqrt{0.5}=\frac1{\sqrt2}$$ and $$\sin\left(\frac\pi4\right)=\frac1{\sqrt2}$$
$\endgroup$ 2 $\begingroup$That's true also for $\cos\left(\frac\pi4\right)$ $$\sin\left(\frac\pi4\right)=\cos\left(\frac\pi4\right)=0.5^{0.5}=\frac1{\sqrt2}=\frac{\sqrt2}{2}$$
Here is a picture of what is going on geometrically:
Since
$$-1\le \sin x \leq 1$$
for each $\theta$ such that $$e^{-\frac 1e} <\sin \theta < 1$$
you can find two values of $x$ such that
$$x^x=\sin \theta$$
and exactly one for $\sin \theta = e^{-\frac 1e}$ and $<\sin \theta = 1$
