Trigonometry in Three Dimensions

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Ordinary trigonomtry is possible wherever there are straight lines. The diagram below shows a rectangular block.

I want ro find the distance AG and the angles AGB and BGD.

To find the distance AG, use Pythagoras Theorem in three dimensions. AB, BC and CG are mutually at right angles, so

Consider the triangle ABG. AB is horizontal and BC and BF are both at right angles to AB, so every line in the plane BFGC is also at right angles to AB. ABG is then a right angled triangle. It is shown below.

The angle AGB is then

To find the angle BGD, construct the triangle BGD, shown below, and find the lengths BD, BG and DG using Pythagoras Theorem three times.

Use the Cosine Rule.