Subsection B.2.3 Important Triangles
¶Computing sine and cosine is non-trivial for general angles — we need Taylor series (or similar tools) to do this. However there are some special angles (usually small integer fractions of \(\pi\)) for which we can use a little geometry to help. Consider the following two triangles.
The first results from cutting a square along its diagonal, while the second is obtained by cutting an equilateral triangle from one corner to the middle of the opposite side. These, together with the angles \(0,\frac{\pi}{2}\) and \(\pi\) give the following table of values
| \(\theta\) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) | \(\csc\theta\) | \(\sec\theta\) | \(\cot\theta\) |
| \(0\) rad | 0 | 1 | 0 | DNE | 1 | DNE |
| \(\tfrac{\pi}{2}\) rad | 1 | 0 | DNE | 1 | DNE | 0 |
| \(\pi\) rad | 0 | -1 | 0 | DNE | -1 | DNE |
| \(\tfrac{\pi}{4}\) rad | \(\tfrac{1}{\sqrt{2}}\) | \(\tfrac{1}{\sqrt{2}}\) | 1 | \(\sqrt{2}\) | \(\sqrt{2}\) | 1 |
| \(\tfrac{\pi}{6}\) rad | \(\tfrac{1}{2}\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{\sqrt{3}}\) | 2 | \(\tfrac{2}{\sqrt{3}}\) | \(\sqrt{3}\) |
| \(\tfrac{\pi}{3}\) rad | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{2}\) | \(\sqrt{3}\) | \(\tfrac{2}{\sqrt{3}}\) | 2 | \(\tfrac{1}{\sqrt{3}}\) |