Since you know the value of sin B and c, you could simply use the equation

sin B=b/c

to calculate b and then plug this into Pythagorean theorem

a^2+b^2=c^2

to get a.

But even simpler: If you know that in right triangle sin A=cos B, then you have

cos B=sin A=a/c

from which you can easily get a.

(I guess that A is the angle opposite to the side a and B is the angle opposite to the side b - this is probably shown in the figure you did not show us. Am I right?)