Nombres complexes
Si \(z = a + ib\), alors \(\rho = |z| = \sqrt{a^2 + b^2}\)
Si \(z = a + ib\), alors \(\theta = \arg(z)\) vérifie \(\cos\theta =\) \(\dfrac{a}{\sqrt{a^2+b^2}}\)
Si \(z = a + ib\), alors \(\theta = \arg(z)\) vérifie \(\sin\theta =\) \(\dfrac{b}{\sqrt{a^2+b^2}}\)
Valeurs trigonométriques
💡 Pour \(\theta\) négatif : \(\sin\) change de signe, \(\cos\) reste identique.
\(\cos 0 =\) \(1\) et \(\sin 0 =\) \(0\)
\(\cos\dfrac{\pi}{6} =\) \(\dfrac{\sqrt{3}}{2}\) et \(\sin\dfrac{\pi}{6} =\) \(\dfrac{1}{2}\)
\(\cos\dfrac{\pi}{4} =\) \(\dfrac{\sqrt{2}}{2}\) et \(\sin\dfrac{\pi}{4} =\) \(\dfrac{\sqrt{2}}{2}\)
\(\cos\dfrac{\pi}{3} =\) \(\dfrac{1}{2}\) et \(\sin\dfrac{\pi}{3} =\) \(\dfrac{\sqrt{3}}{2}\)
\(\cos\dfrac{\pi}{2} =\) \(0\) et \(\sin\dfrac{\pi}{2} =\) \(1\)
\(\cos\dfrac{2\pi}{3} =\) \(-\dfrac{1}{2}\) et \(\sin\dfrac{2\pi}{3} =\) \(\dfrac{\sqrt{3}}{2}\)
\(\cos\dfrac{3\pi}{4} =\) \(-\dfrac{\sqrt{2}}{2}\) et \(\sin\dfrac{3\pi}{4} =\) \(\dfrac{\sqrt{2}}{2}\)
\(\cos\dfrac{5\pi}{6} =\) \(-\dfrac{\sqrt{3}}{2}\) et \(\sin\dfrac{5\pi}{6} =\) \(\dfrac{1}{2}\)
\(\cos\pi =\) \(-1\) et \(\sin\pi =\) \(0\)
\(\cos\dfrac{-\pi}{6} =\) \(\dfrac{\sqrt{3}}{2}\) et \(\sin\dfrac{-\pi}{6} =\) \(-\dfrac{1}{2}\)
\(\cos\dfrac{-\pi}{4} =\) \(\dfrac{\sqrt{2}}{2}\) et \(\sin\dfrac{-\pi}{4} =\) \(-\dfrac{\sqrt{2}}{2}\)
\(\cos\dfrac{-\pi}{3} =\) \(\dfrac{1}{2}\) et \(\sin\dfrac{-\pi}{3} =\) \(-\dfrac{\sqrt{3}}{2}\)
\(\cos\dfrac{-\pi}{2} =\) \(0\) et \(\sin\dfrac{-\pi}{2} =\) \(-1\)
\(\cos\dfrac{-2\pi}{3} =\) \(-\dfrac{1}{2}\) et \(\sin\dfrac{-2\pi}{3} =\) \(-\dfrac{\sqrt{3}}{2}\)
\(\cos\dfrac{-3\pi}{4} =\) \(-\dfrac{\sqrt{2}}{2}\) et \(\sin\dfrac{-3\pi}{4} =\) \(-\dfrac{\sqrt{2}}{2}\)
\(\cos\dfrac{-5\pi}{6} =\) \(-\dfrac{\sqrt{3}}{2}\) et \(\sin\dfrac{-5\pi}{6} =\) \(-\dfrac{1}{2}\)
Combinatoire
\(A^n_p =\) \(\dfrac{n!}{(n-p)!}\)
\(C^n_p =\) \(\dfrac{n!}{p!\,(n-p)!}\)
Lois de probabilité
Si \(X \sim \mathscr{B}(n,p)\), alors \(X(\Omega) =\) \([\![0\,;\,n]\!]\)
Si \(X \sim \mathscr{B}(n,p)\), alors \(p(X\!=\!k) =\) \(C_n^k\,p^k\,(1-p)^{n-k}\)
Si \(X \sim \mathscr{B}(n,p)\), alors \(E(X) =\) \(np\)
Si \(X \sim \mathscr{B}(n,p)\), alors \(V(X) =\) \(np(1-p)\)
Si \(X \sim \mathscr{H}(N,n,p)\), alors \(\min(X(\Omega)) =\) \(\max\{0\,;\, n-(1-p)N\}\)
Si \(X \sim \mathscr{H}(N,n,p)\), alors \(\max(X(\Omega)) =\) \(\min\{n\,;\, pN\}\)
Si \(X \sim \mathscr{H}(N,n,p)\), alors \(p(X\!=\!k) =\) \(\dfrac{C_{Np}^k\,C_{N-Np}^{n-k}}{C_N^n}\)
Si \(X \sim \mathscr{H}(N,n,p)\), alors \(E(X) =\) \(np\)
Si \(X \sim \mathscr{H}(N,n,p)\), alors \(V(X) =\) \(np(1-p)\,\dfrac{N-n}{N-1}\)
Si \(X \sim \mathscr{G}(p)\), alors \(X(\Omega) =\) \(\mathbb{N}^*\)
Si \(X \sim \mathscr{G}(p)\), alors \(p(X\!=\!k) =\) \((1-p)^{k-1}\,p\)
Si \(X \sim \mathscr{G}(p)\), alors \(E(X) =\) \(\dfrac{1}{p}\)
Si \(X \sim \mathscr{G}(p)\), alors \(V(X) =\) \(\dfrac{1-p}{p^2}\)
Si \(X \sim \mathscr{P}(\lambda)\), alors \(X(\Omega) =\) \(\mathbb{N}^*\)
Si \(X \sim \mathscr{P}(\lambda)\), alors \(p(X\!=\!k) =\) \(e^{-\lambda}\,\dfrac{\lambda^k}{k!}\)
Si \(X \sim \mathscr{P}(\lambda)\), alors \(E(X) =\) \(\lambda\)
Si \(X \sim \mathscr{P}(\lambda)\), alors \(V(X) =\) \(\lambda\)
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