Appendix 3 - for Advanced Readers

More About Irrational Numbers

Except in the very special case where we are given enough information, for example three sides equal, or two sides equal and one angle of 60°, to know that a triangle is equilateral, every time we use the Sine and Cosine Rules to solve a triangle at least one of the sides or angles we find will be irrational. We only need *three* pieces of information about a triangle to be able to solve it; if we are given *four* pieces we can solve it in four different ways and because of rounding errors inside your calculator when using irrational numbers they will not necessarily all give exactly the same result. In mathematics we must always be careful when we are given *redundant* information.

From this it follows that, if we are solving a triangle when we are given three pieces of information as rational numbers, at least one of the other three quantities to be found will be irrational, so we shall never find an angle of exactly 90° except in a triangle where one of the angles is 30° or 60° and the longest side is twice the length of the shortest, in which case the third side is irrational.