Insulating the underside of the ceiling between rafters
We have a one-story house with a near flat roof close to the beach in Southern California. Winter temps drop into the 40s. Our ceilings are pushed up to the rafters, so what you see inside are the 4x6 beams with 1x8 shiplap boards running across the tops. On top of the 1x8s is the roofing material.
The ceilings get quite warm in the summer, but that isn't really a problem because opening the windows will bring in plenty of cooler air. The problem is during the winter time. We'd like to insulate the ceiling but will not do so if it means covering the wood beams with drywall.
Our idea is to put 1" rigid foam insulation between the rafters up against the shiplap boards. Then we'd cover the insulation with 1/2" drywall, leaving us with 4" of beam exposed. It won't look as nice as the current ceiling but if it means the house will be warmer in the winter, we'd do it.
Our questions are:
1) Will an inch of rigid foam insulation do anything?
2) Is it okay to have the insulation pressed against the underside of the roof?
3) Can you think of any alternative solutions that would provide some insulation while preserving the exposed wood beams?
Thanks so much for your help.