Right triangle resolution using trigonometric identities
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.
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, $$c=\dfrac{6}{\tan(30^\circ)}=\dfrac{6}{\dfrac{\sqrt{3}}{3}}=\dfrac{18}{\sqrt{3}}$$
Thus, the solution is:
$$C=60^\circ \qquad a=12 \qquad c=\dfrac{18}{\sqrt{3}}$$