How do I calculate the required heading for an aircraft carrier with an angled deck to sail into the wind? The natural wind is from 360 degrees at 10 knots. The desired relative wind down the angled deck is 10 degrees left of the ship’s heading at 30 knots. 

What heading and speed must the carrier go?
A carrier heading of 010 degrees at a speed of 20 knots would give you an acceptable solution.  You could perform some vector addition to compute the resultant wind and perhaps optimize this somewhat, but in reality the winds are generally not constant enough to make minor adjustments worth chasing.  

Even if the winds were calm and the ship was making all 30 knots the crosswind component is almost negligible, and well within limits for any carrier based aircraft.  This happens a lot.  There is a bigger negative effect from  turbulence caused by the island structure when the ship has to make wind than any effect of increased crosswind.