Formules de trigonométrie - Forme exponentielle

\(\cos \dfrac \pi 4 = \cos \left(2\times \dfrac \pi 8\right) = 2\cos^2\dfrac \pi 8 - 1\)

on peut donc isoler \(\cos \dfrac \pi 8\):

\(cos^2 \dfrac \pi 8 = \dfrac 1 2 \left(1+\cos \dfrac \pi 4 \right) = \dfrac 1 2 \left(1+\dfrac {\sqrt 2} 2 \right) = \dfrac {2+\sqrt 2} 4\)

Or \(\cos \dfrac \pi 8>0\) donc \(\cos \dfrac \pi 8 = \sqrt{\dfrac {2+\sqrt 2} 4}\)

\(\sin^2 \dfrac {\pi} 8 = 1-cos^2 \dfrac \pi 8 = 1-\dfrac {2+\sqrt 2} 4\)

Or \(\sin \dfrac \pi 8>0\) donc \(\sin \dfrac \pi 8 = \sqrt{\dfrac {2-\sqrt 2} 4}\)