TITLE: Calculate $x$, if $y = a \cdot \sin{[b(x-c)]}+d$
QUESTION [5 upvotes]: I am not an expert when it comes to trigonometric functions.
I need to calculate value of $x$ for a program.
If $y = a \cdot \sin{[b(x-c)]}+d$ then what will be the formula to calculate $x$?
Please help..

REPLY [3 votes]: It all boils down to the following:
If two real quantities $t$ and $s$ are related by the equation $$(*) \qquad s=\sin t$$ then for arbitrary $t\in{\mathbb R}$ the corresponding $s$ is uniquely determined and can, e.g., be evaluated by means of the $\sin$-series. The $s$-values appearing in this way turn out to be restricted to the interval $[-1,1]$.
The other way around the situation is more complicated. For a given $s\in\ [-1,1]$ there is a unique $T\in [-{\pi\over2},{\pi\over2}]$ that satisfies the equation $(*)$, and this particular $T$ is called $\ \arcsin s$. All other $t$ that satisfy $(*)$ can be obtained from this T, and by looking at the graph of the $\sin$-function one gets the following list of values: 
If $s=-1$ then $T=-{\pi/2}$, and if $s=1$ then $T={\pi/2}$. In both these cases the complete set of $t$ satisfying $(*)$ is given by $\{T+2k\pi\ |\ k\in{\mathbb Z}\}$. If $s$ lies strictly between $-1$ and $1$ then each $t$-interval of length $2\pi$ contains ${\it two}$ points that satisfy $(*)$, and the complete set of such $t$ is given by $\{T+2k\pi\ |\ k\in{\mathbb Z}\}\ \cup\ \{(2k+1)\pi-T\ |\ k\in{\mathbb Z}\}$.