Question

Assume angle C is a right angle and given the conditions b=2,angle A=30 solve for the side lengths of the right triangle.

Answer 1

If C is a right triangle and angle A is 30 degrees, then angle B is 60 degrees.
Thus, we have a 30-60-90 triangle.

You ask for the side lengths of the right triangle, but you provide the value 2 as the length of b.  Thus, your goals are to find the side lengths a and c.

Using the Law of Sines, 

    a           b            c
-------- = -------- = ---------
sin A       sin B      sin C

Using the info given, 

     a               2              c
--------- = ----------- = -------------
sin 30       sin 60        sin 90

Using the last two ratios, 2 sin 90 = c sin 60, or 2 = c sin 60, or c = 2/sin 60, or c = 2/(sqrt(3)/2) = sqrt(3)

Then c = sqrt(3), c = 2 and a = 1.

Answer 2

A = 2*tan(30°) = 2(√3)/3 
b = 2 … given 
c = 2/cos(30°) = 4(√3)/3