I've just covered Taylor polynomials and series theory, but there are precious few examples so i'm trying to do the following exercise:

Compute Taylor series of f(x) = cos 3x at pi/12

My attempted answer:

f(x)= cos3x = cos pi/4

f'(x) = -3 sin 3x = -3 sin pi/4

f''(x) = -9 cos 3x = -9 cos pi/4

etc

so Tn cos3x = cos3x + (-3 sin 3x) x/2! + (-9 cos pi/4) x^2/ 3!...

I can't figure how I should go about formulating a series formula for this , especially as trig functions are involved. I'm also not even sure I'm doing the Taylor formula right as i've seen some wierd trig series. Any advice on how to start would be much appreciated

Thanks