Is there a way to guess the trigonometric angle(radian) at which a particular fractional number is obtained? While solving problems, I encounter fractional numbers like 1/250 for which I need to find the angle(radian) of Sine at which this fractional number 1/250 is obtained. So, is there a way to make an approximation of the required angle? Answers much appreciated!

The arc sine of small angles ($|x|<1$) can be computed using the Taylor development around $0$:

$$\arcsin(x)=x+\frac{1\cdot3}{2\cdot4}\frac{x^3}3+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^5}5+\cdots.$$

$$\arcsin\frac1{250}\approx0.004+0.000000008+0.000000000000064+\cdots$$

For very small arguments or low accuracy requirements, the first term can be enough.