84 lines
1.5 KiB
TeX
84 lines
1.5 KiB
TeX
\langchapter{Trigonométrie}{Trigonometry}
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%TODO Complete chapter
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\langsection{Cercle unitaire}{Unit circle}
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%TODO Complete section
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Le cercle unitaire est un cercle de centre $(0,0)$ et de rayon 1.
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\subsection{cos}
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%TODO Complete subsection
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$\cos 0 = 1$
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$\cos \frac{\pi}{2} = 0$
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$\cos \pi = -1$
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$\cos(-\frac{\pi}{2}) = 0$
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$\cos(\pi + t) = -\cos(t)$
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$\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$
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$\cos\frac{\pi}{3} = \frac{1}{2}$
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$\forall (a,b) \in \R^2$
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$\cos(a + b) = \cos a \cos b + \sin a \sin b$
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$\cos(a - b) = \cos a \cos b - \sin a \sin b$
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$\cos a + \cos b = 2 \cos(\frac{a + b}{2}) \cos(\frac{a - b}{2} )$
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\subsection{sin}
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%TODO Complete subsection
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$\sin 0 = 0$
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$\sin(\pi - t) = \sin(t)$
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$\sin(\frac{\pi}{2} - t) = \cos(t)$
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$\sin \frac{\pi}{6} = \frac{1}{2}$
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$\sin \frac{\pi}{2} = 1$
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%$\sin(\frac{\pi}{2} + t) = -\cos(t)$
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$\forall (a,b) \in \R^2$
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$\sin(a + b) = \sin a \cos b + \sin b \cos a$
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$\sin(a - b) = \sin a \cos b - \sin b \cos a$
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$\sin a - \sin b = 2 \cos (\frac{a+b}{2}) \sin (\frac{a-b}{2})$
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$\sin a\sin b = \frac{\cos(a - b) - \cos(a + b)}{2}$
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\subsection{tan}
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%TODO Complete subsection
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$\tan 0 = 0$
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$\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$
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$\tan \frac{\pi}{4} = 1$
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$\tan(\frac{\pi}{2} - x) = \frac{1}{\tan x}$
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$\tan(\frac{\pi}{2} + x) = -\frac{1}{\tan x}$
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$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1- \tan(a)\tan(b)}$
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$\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$
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\subsection{Combinaisons}
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%TODO Complete subsection
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$\forall (a,b) \in \R^2$
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$\sin a \cos b = \frac{\sin(a + b) + \sin(a - b)}{2}$
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