|you see that, keeping full contact along the arc, you get the most bend in the paper at 90 degrees (cosign of 90 = 1). For lesser angles the arc is less. The maximum arc you can produce (at 90 degrees) is the turn radius.
Si--you'd better stick to your day job. (Oh--wait, you ARE a scientist!)
The cosine (not "cosign") of 90 is zero. Yes, you do get the maximum bend (minimum radius) as the ski approaches 90 degrees edge angle. But intuitively, with your "paper ski," you can see that at 90 degrees, you cannot make full contact with the "ski's" edge. You could fold the ski right double--radius of arc=zero--and you STILL couldn't make full contact with its edge!
Furthermore, the higher the edge angle, the greater the proportion of the skier's force against the ski goes into bending it, with less pressure forcing it "down" into the snow. At 90 degrees, there will be little-to-no "downward" pressure on the snow. (The only reason there would be any at all is if the ski is tipped more than 90 degrees to the "line of action"--in other words, the skier is angulated.)
You are right, though, that the ski will bend into a tighter arc (on a hard surface), the higher the edge angle, as most skiers have experienced, and as Physicsman's formula shows. As the edge angle decreases, the radius will approach the ski's built-in sidecut radius, as others have described. The LONGEST radius a ski could theoretically carve (pure carve) is its sidecut radius.