Convert sin calculation to use per-frame delta - math

Question

I currently have a function that updates a value using the sin function, and a timeFactor double that keeps track of how much time has passed since the program started:
double timeFactor;
double delta;
while(running) {
delta = currentTime - lastTime;
timeFactor += delta;
var objectX = sin(timeFactor);
}
However, I need to convert this code to use the delta rather than the timeFactor.
I.e. For updating to sin(time+delta) I only want to use the current value of sin(time) and anything calculated from the value of delta.
I.e. calcualte sin(time+delta) == f(sin(time),delta)
How do I do this?

Answer 1

From math:
sin(A + B) == sin(A) * cos(B) + cos(A) * sin(B)
cos(A + B) == cos(A) * cos(B) - sin(A) * sin(B)
Store sin(A) and cos(A) in two variables.
Then for updating them use temporary copy of one of them, otherwise you will update the second using the new instead of the old value of the first.
Assuming:
persistent objectX stores current and is initialised with initial sin(timeFactor)
persistent objectXc stores current and is initialised with initial cos(timeFactor)
temporary objectXt stores a copy of objectX
("persistent" as in "keeps value across executions of update code",
in contrast to "temporary" as in "only keeps value during update code";
this is to avoid using the "global" attribute, which implies poor data design)
Update code:
objectXt = objectX;
objectX = objectX * cos(delta) + objectXc * sin(delta);
objectXc = objectXc* cos(delta) - objectXt * sin(delta);
Credits to John Coleman for spotting the problem in initial idea to use
1 == sin(A)*sin(A)+cos(A)*cos(A)
That would have been actually
sin(time+delta)== f(sin(time), delta)
But it fails for 50% of a full period.
So I hope this
sin(time+delta)==f(sin(time), cos(time), delta)
also helps.