1.82: Trigonometric identities

A pyramid with a rectangular base $ABCD$ and vertex $E$ is shown in Figure 6.

The rectangular base is horizontal with $AB=12\text{ cm}$ and $BC=8\text{ cm}$. The diagonals of the base intersect at the point $O$. The vertex $E$ of the pyramid is vertically above $O$. The height of the pyramid is $h$ cm and $AE=BE=CE=DE=10\text{ cm}$.

(a) Show that $h=4\sqrt3$. (3)

(b) Find, in degrees to 1 decimal place, the size of angle $OCE$. (2)

The angle between $OE$ and the plane $CBE$ is $\theta^\circ$.

(c) Show that $\cos\theta^\circ=\dfrac{2\sqrt7}7$. (3)

The point $P$ is the midpoint of $BE$ and the point $Q$ is the midpoint of $CE$.

(d) Find, in degrees to 1 decimal place, the size of the angle between the plane $OPQ$ and the plane $EPQ$. (4)