Convergence of the Taylor series for the sine function

The Taylor series e.g. for sin(x) (and other formulas involving trigonometric functions) work with radians instead of degrees because they were formulated to work with radians. If x is not measured in radians, the expression sin(x)=x-x^3/3!+x^5/5!... doesn't make much sense. (What does it mean to add different powers of the unit degree? Not much.) If one uses the right forms of the derivatives, which have the unit 1/degree, then the degrees disappear in the formula, and sin is dimensionless.

Will it converge? Yes. Will it converge to the desired answer? No. Think of how you get a Taylor series, by repeatedly taking a derivative. The derivative of sin x is only cos x if x is in radians. If x is in degrees, then the derivative if pi/180 cos x. Using this you could create the correct Taylor series if you really wanted to.