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Subsection B.1.2 Pythagoras

Since trigonometry, at its core, is the study of lengths and angles in right-angled triangles, we must include a result you all know well, but likely do not know how to prove.

The lengths of the sides of any right-angled triangle are related by the famous result due to Pythagoras

\begin{align*} c^2 &= a^2+b^2. \end{align*}

There are many ways to prove this, but we can do so quite simply by studying the following diagram:

We start with a right-angled triangle with sides labeled \(a,b\) and \(c\text{.}\) Then we construct a square of side-length \(a+b\) and draw inside it 4 copies of the triangle arranged as shown in the centre of the above figure. The area in white is then \(a^2+b^2\text{.}\) Now move the triangles around to create the arrangement shown on the right of the above figure. The area in white is bounded by a square of side-length \(c\) and so its area is \(c^2\text{.}\) The area of the outer square didn't change when the triangles were moved, nor did the area of the triangles, so the white area cannot have changed either. This proves \(a^2+b^2=c^2\text{.}\)