In a plane triangle if SinA / Sin B = 2/3 then what is angle ratio (A / B)?

Let's look at some limiting cases. When sin(B) = 1 and sin(A) = 2/3, then

A/B = 41.81031 deg/90 deg = 0.46456.

On the other hand, when A and B are both very small,

sin(A)/sin(B) is nearly A/B, so A/B = about 0.6666... .

Perhaps what is meant is that it's a RIGHT triangle, which would mean that A + B = 90 degrees, and at the same time, sin(A)/sin(B) = 2/3. Then sin(A)/cos(A) = 2/3, so tan(A) = 2/3 and A = 33.69 deg, B = 56.31 deg, and A/B = 0.5983.

If the latter answer is what was wanted, your teacher is a MORON. He or she should have said "right triangle," not "plane triangle."

I suggest you try it on your own.