calculus is a technique that applies to any functions. it does not USE trig, rather if you know calculus you can apply it to understand trig functions better.
there are several important classes of functions to which we want toa pplythe ideas of calculus. these are usually polynomials, trig functions, and exponential and log functions.
calculus is very easy to apply to, polynomials, indeed one does not to understand limits at all to apply calculus to them, everything can be done by algebra, musing the explicit formulas they have.
the first inetersting, i.e. mroe difficult functions we try to apply calculus to, are trig functions, because they do not have simple formulas. In fact trig functions cannot even be rigorously defined without suing calculus, because sin and cosine are actually inverse functions of arclength, which requires calculus for its very definition.
after applying the ideas of calculus to trig, a wonderful thing happens: after the fact, one can actualkly write down simple, but infinite, formulas for the trig functions and hence understand them much better than before.
trig functions are important because there are circles in the world which we want to straighten out and measure. I.e. they are relevant to measuring circular arc length.
calculus is useful aprtly becaue it helps us understand important functions, like trig functions and exponential functions, which are even harder to define without calculus.
i.e. the best way to define natural log, is as an area function, for which again calculus is needed.
