a. If you have a rectangular triangle and an angle of 30°,it means that the opposite side of the 30° angle is 1/2 of hypotenuse.
Let hypotenuse be b.
=> a=b/2
From the Pythagoras => b² = a² +3²
(2a)² = a² + 9
4a² = a² +9
3a² = 9 ⇒ a²=1/3 ⇒ a=1/√3 ⇒ a=√3/3
b. a=b/2 ⇒ b=2√3/3
c. In the triangle we have an angle of 90° and one of 30°
↓
θ = 180°-90°-30°
θ = 60°
cos θ = cos 60° = 1/2