Derivation of formula for general formula of sine equations

[autodidude]

My book gives the following formulas for the general solution of sine equations (if sin(x)=a)

$$x=2n\pi + sin^-^1(a)$$

and

$$x=(2n+1)\pi - sin^-^1(a)$$

Alternatively
$$x=n\pi + (-1)^nsin^-^1(a)$$

But it doesn't explain how it got them. I can easily see how they got the first two but I have no idea how they got the more general one. My teacher didn't show how it was derived either...just threw a bunch of formulas at us to remember

 

[micromass]

Take the equation

$$x=n\pi + (-1)^n \sin^{-1}(a)$$

What happens if you take n even (that is: of the form 2k) and what happens if you take n odd (that is: of the form 2k+1)??

 

[autodidude]

^ Wow, it seems so obvious now but I never would've seen had you not pointed me in the right direction, thank you!