Решение:

1. Если $$sin \alpha = \frac{1}{3}$$, то:

   a) $$cos^2 \alpha + sin^2 \alpha = 1$$, следовательно, $$cos \alpha = \pm \sqrt{1 - sin^2 \alpha} = \pm \sqrt{1 - \frac{1}{9}} = \pm \sqrt{\frac{8}{9}} = \pm \frac{2\sqrt{2}}{3}$$

   б) $$tg \alpha = \frac{sin \alpha}{cos \alpha} = \frac{\frac{1}{3}}{\pm \frac{2\sqrt{2}}{3}} = \pm \frac{1}{2\sqrt{2}} = \pm \frac{\sqrt{2}}{4}$$

2. Если $$cos \alpha = \frac{1}{2}$$, $$sin \alpha = \frac{\sqrt{3}}{2}$$, то:

   а) Уже дано, $$cos \alpha = \frac{1}{2}$$

   б) $$tg \alpha = \frac{sin \alpha}{cos \alpha} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

3. Найти: $$sin 150^\circ$$, $$cos 135^\circ$$

   $$sin 150^\circ = sin (180^\circ - 30^\circ) = sin 30^\circ = \frac{1}{2}$$

   $$cos 135^\circ = cos (180^\circ - 45^\circ) = -cos 45^\circ = -\frac{\sqrt{2}}{2}$$