Given a right triangle with one of the cathetus $$b = 6$$ cm and the angle $$B = 30^\circ$$. find the sides and the unknown angle.
See development and solution
Development:
In this case we know one of the cathetus and one of the angles, therefore we can follow fourth case. As such:
The angle $$C$$ is, $$$C=90^\circ -30^\circ=60^\circ$$$
The side $$a$$ is, $$$a=\dfrac{6}{\sin(30^\circ)}=\dfrac{6}{\dfrac{1}{2}}=12$$$
The side $$c$$ is, $$$c=\dfrac{6}{\tan(30^\circ)}=\dfrac{6}{\dfrac{\sqrt{3}}{3}}=\dfrac{18}{\sqrt{3}}$$$
Solution:
Thus, the solution is:
$$$C=60^\circ \qquad a=12 \qquad c=\dfrac{18}{\sqrt{3}}$$$