Take a look at Ker-I Ko's book or his survey. I don't have the book with me right now. You can find the following in Weirauch's book:
"On compact subspaces of their domains real functions exp, sin, cos, tan, and their inverse can be computed in time $t_m(k)\log(k)$."
Where $t_m(k)$ is the time needed to compute the first k digits of the multiplication (of two real numbers) which is $O(k^2)$. This is from page 229. In the introduction he mentions that multiplication, $\sin$, $\exp$, and $\log$ can be computed in time $O(k^2)$.
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Ker-I Ko, "Polynomial-time computability in analysis", in "Handbook of Recursive Mathematics," Volume 2, Recursive Algebra, Analysis and Combinatorics, Yu. L. Ershov et al., Eds., 1998, pp. 1271-1317. http://www.cs.sunysb.edu/~keriko/survey3.ps
Ker-I Ko, "Computational Complexity of Real Functions", Birkhauser, 1991
Klaus Weihrauch, "Computable Analysis: An Introduction", (Texts in Theoretical Computer Science. An EATCS Series), Springer, 2000
