The 'exact' answer for the angle you need to row (alpha) is Sin(alpha)=KCos(Phi).

Yes, the 'exact' answer is the tangent. If you want the relationship between K and Phi, then:

K = Cos(Phi) + SQR( (pi+phi)^2 - sin^2(phi))

The derivation of both of these is contained in the article.