Okay, so heres what I mean by this... I know that:

[a]

and

[b]

So, with that, I also know that:

and

Now, is there some way to continue this process? To find some forumula for

simmilar to the forumulas above? And then use the derivitive of this sin function to find the forumula for the corresponding formula

? Like, for the examples above, and

Is there a general formula for this that we can use by differentiating it to get to the cosine version of said formula? Or is that not a valid method to use to get from the sine version to the consine version?