To be clearer, $45^{\circ} = \frac{\pi}{4}$ radians which estimates to $0.78539816$ radians; whereas, $\sin(45^{\circ}) = \frac{\sqrt2}{2}$ estimates to $0.70710678$ which is a ratio and has no units.
Why does $\sin(0.78539816) = 0.70710678$? Or why does the $\arcsin(0.70710678) = 0.78539816$ radians?
I want to be able to convert ratios of sides to angles without a trigonometric calculator or trigonometric tables.
I´d like to be able to go from knowing that $\sin(45^{\circ}) = \frac{\sqrt2}{2}$and then be able to tell how many radians and degrees it is without a trigonometric calculator or trigonometric tables. Is this possible?
Or considering the 3-4-5 right triangle, I´d like to be able to know that the $\arcsin(\frac{4}{5}) = 0.92729522$ radians or $53.13010235^{\circ}$ without a trigonometric calculator or trigonometric tables. I understand how to convert between radians and degrees using $\pi = 180^{\circ}$ , but I don´t understand how to go from $\arcsin(\frac{4}{5})$ to $0.92729522$ radians.
I was thinking that it related to the domain $[-1,1]$ of the sin function, but radians are the range units on the sine wave curve. Is it somehow related to polar coordinates where $x=r*\cos(θ)$ and $y=r*\sin(θ)$?
Also, this website seems to hint at an extension of the radius to calculate tangent so might that be the difference in the numbers? Link