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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 \(\tfrac{1}{\sqrt{3}}\)