Since the surreal numbers form a saturated real-closed field, it follows that they serve as a (proper class) version of the hyperreal numbers, including the [transfer principle](https://en.wikipedia.org/wiki/Hyperreal_number#The_transfer_principle). This means that absolutely every function and relation on the real numbers extends to the surreal field, with furthermore all the same first-order expressible properties. 
 $$\langle\mathbb{R},+,\cdot,0,1,<,\mathbb{Z},\sin,\exp,\ldots\rangle\prec\langle\text{No},+,\cdot,0,1,<,\mathbb{Z}^*,\sin^*,\exp^*,\ldots\rangle$$
So we will have a surreal sine function, cosine, exponential, logarithm, and so forth---every function whatsoever has a surreal analogue with the same first-order expressible features in $\text{No}$ that the original function has in $\mathbb{R}$.

In particular, yes, there will be a periodic nonstandard function $\cos^*$, with all the same properties of the $\cos$ function on the reals---it will be periodic with period $2\pi$ and obey the expected identities with other trigonometric functions, $\sin^{2}(x)+\cos^{2}(x)=1$. Regarding the power series representation, the nonstandard power series for $\sin^*$ will have terms extending into $\mathbb{N}^*$, which is a proper class, and so there will be various subtle set-theoretic issues to engage with.

The nonstandard analogues $\sin^*$, $\exp^*$, however, are definitely not unique except in trivial cases. One can see this immediately because the surreal field has a superabundance of automorphisms, and there is nothing special about $\omega$, for example, in the field structure alone, for it can be moved by field automorphisms to any other infinite element. In light of this, there will be many distinct versions of the nonstandard functions, with all the same expressible properties. In this sense, it is not directly sensible to ask, what is the value of $\cos^*(\omega)$? In fact there are versions of $\cos^*$ for which this is $0$, or $1$, or $-1/\sqrt{2}$, and so forth, simply because of the automorphisms.

Therefore, when asking for surreal versions of the various standard functions, one should specify more exactly what is desired. In particular, how do you want the function to interact with the genetic birthday structure? The birthday structure is not determined by the algebraic surreal structure. The surreals are not saturated with respect to the birthday structure, and the field is rigid when one incorporates that structure as a part of what is meant by the surreals.

For this, there may be some good accounts of which $\sin^*$ to use, and I shall leave this for others.