The 30° and 60° Points

Half an equilateral triangle is a 30-60-90 triangle. With hypotenuse 1, the side opposite 30° is half of it, 1/2, and Pythagoras gives the other leg: $\sqrt{1-\frac{1}{4}} = \frac{\sqrt{3}}{2}$.

At 30° the long leg is the across and the short leg the height, so the point is ($\frac{\sqrt{3}}{2}$, $\frac{1}{2}$): $\cos\left(30^{\circ}\right) = \frac{\sqrt{3}}{2}$ and $\sin\left(30^{\circ}\right) = \frac{1}{2}$.

Stand the same triangle on its other end and the 60° angle is at the centre. Which leg is which is then the whole question: the leg opposite the angle at the centre is the height.