Quick questions on Three-dimensional trigonometry for HSC Maths Advanced: finding the right-angled triangle inside a rectangular box or pyramid, the space diagonal, the angle between a line and a plane, the height of an apex above the base centre, and problems that combine a base-plane Pythagoras result with a vertical rise

A solid is a collection of points, edges and flat faces. Trigonometry only ever acts on a single flat triangle, so the first move is always to isolate a triangle that lies in one plane and is right-angled. Two facts generate almost every such triangle in Year 11 work:

A rectangular prism (box) with edge lengths $l$, $w$ and $h$ has two kinds of diagonal, and both come from the same idea, applied once or twice.

For a right pyramid on a square base, the apex sits directly above the centre of the base, and the vertical height drops from the apex to that centre, meeting the base at a right angle. This single fact unlocks the pyramid, because it ties the apex to the one special point from which the base diagonals are easy.

The hardest Year 11 three-dimensional questions never leave the toolkit; they just need two triangles in different planes, solved in order. The pattern is: solve a triangle lying in the base (often by Pythagoras) to get a length you do not start with, then carry that length into a second, vertical triangle to get the angle or height asked for. The elevation-from-two-positions problem is the classic case where both sightings share one vertical plane and the unknowns are found by setting two tangent equations equal. The worked examples below drill exactly these chains.

As with all degree-measure trig, set the calculator to DEG and confirm $\sin 30\degree = 0.5$ before solving.