Mathématique – Formules

Algèbre :

\[(a+b)^2=a^2+2\mathit{ab}+b^2\]

\[(a-b)^2=a^2-2\mathit{ab}+b^2\]

\[a^3-b^3=(a-b)(a^2+\mathit{ab}+b^2)\]

\[(a+b)^3=a^3+3a^2b+3\mathit{ab}^2+b^3\]

\[(a-b)^3=a^3-3a^2b+3\mathit{ab}^2-b^3\]

\[a^3+b^3=(a+b)(a^2-\mathit{ab}+b^2)\]

\[a^3-b^3=(a-b)(a^2+\mathit{ab}+b^2)\]

\[\frac{-b\pm \sqrt{b^2-4\mathit{ac}}} 2a\mathit{pour}\mathit{trouver}\mathit{les}\mathit{racines}\mathit{de}\mathit{ax}^2+\mathit{bx}+c=0\]

Les nombres complexes :

\[\mathit{Forme}\mathit{alg\text{é}brique}:z=a+\mathit{bi}\]

\(\left|z\right|=\sqrt{a^2+b^2}\mathit{avec}\left|z\right|\mathit{repr\text{é}sente}\mathit{module}\) \(\cos \phi =\frac a{\left|z\right|}\mathit{et}\sin \phi =\frac b{\left|z\right|}\mathit{pour}\mathit{trouver}\phi \mathit{et}\phi \mathit{repr\text{é}sente}l'\mathit{argument}\)

\[\mathit{Forme}\mathit{trigonom\text{é}trique}:z=r\mathit{cis}\phi\]

\(\mathit{Pour}\mathit{tout}\mathit{naturel}\mathit{non}\mathit{nul}n,\mathit{on}a:(\mathit{cis}\phi )^n=\mathit{cis}(n\phi )\) \(\mathit{Pour}\mathit{tout}\mathit{naturel}\mathit{non}\mathit{nul}n,\mathit{on}a:(r\mathit{cis}\phi )^n=r^n\mathit{cis}(n\phi )\)

\[\mathit{Si}z=r\mathit{cis}\phi \mathit{Alors}\mathit{conjugu\text{é}}\mathit{de}\text z=r\mathit{cis}(-\phi )\]

\[\mathit{Si}z=r\mathit{cis}\phi \mathit{Alors}\frac 1 z=\frac 1 r\mathit{cis}(-\phi )\]

\[\mathit{si}z_1=r_1\mathit{cis}\phi _1\mathit{et}z_2=r_2\mathit{cis}\phi _2\mathit{alors}z_1\ast z_2=r_1\ast r_2\mathit{cis}(\phi _1+\phi _2)\]

\[\mathit{si}z_1=r_1\mathit{cis}\phi _1\mathit{et}z_2=r_2\mathit{cis}\phi _2\mathit{alors}\frac{z_1}{z_2}=\frac{r_1}{r_2}\mathit{cis}(\phi _1-\phi _2)\]

Les fonctions cyclométriques

\(\arcsin x\in \frac{-\pi } 2_{\mathit{compris}};\frac{\pi } 2_{\mathit{compris}}\mathit{et}\mathit{avec}x\in -1_{\mathit{compris}};1_{\mathit{compris}}\) \(\arccos x\in 0_{\mathit{compris}};\pi _{\mathit{compris}}\mathit{et}\mathit{avec}x\in -1_{\mathit{compris}};1_{\mathit{compris}}\)

\[\arctan x\in \mathbb{R}\mathit{et}\mathit{avec}x\in \frac{-\pi } 2_{\mathit{compris}};\frac{\pi } 2_{\mathit{compris}}\]

\(\arccot x\in \mathbb{R}\mathit{et}\mathit{avec}x\in \frac{-\pi } 2_{\mathit{compris}};\frac{\pi } 2_{\mathit{compris}}\)

1.8043in \((\arcsin x)'=\frac 1{\sqrt{1-x^2}}\) \((\arccos x)'=\frac{-1}{\sqrt{1-x^2}}\) \((\arctan x)'=\frac 1{1+x^2}\) \((\arccot x)'=\frac{-1}{1+x^2}\)

2.4209in \((\arcsin x)'=\frac 1{\sqrt{1-x^2}}\) \((\arccos u(x))'=\frac{-u'(x)}{\sqrt{1-u^2(x)}}\) \((\arctan x)'=\frac 1{1+x^2}\) \((\arccot u(x))'=\frac{-u'(x)}{1+u^2(x)}\)

1.8043in :math:` ` \((\arccos x)'=\frac{-1}{\sqrt{1-x^2}}\) \((\arctan x)'=\frac 1{1+x^2}\) \((\arccot x)'=\frac{-1}{1+x^2}\)

Les fonctions exponentielles et logarithmes

\(a^x\ast a^y=a^{x+y}\)     \(e^x\ast e^y=e^{x+y}\)

\((a^x)^y=a^{\mathit{xy}}\)       \((e^x)^y=e^{\mathit{xy}}\)

\((a^x)'=a^x\ln a\)     \((e^x)'=e^x\)

\((a^{u(x)})'=\ln a(u'(x))a^{u(x)}\) \((e^{u(x)})'=(u'(x))e^{u(x)}\)

\[a^x=e^{x\ln a}\]

\[\log _ax=\frac{\ln x}{\ln a}\]

\(\ln a+\ln b=\ln (\mathit{ab})\) \(\log _ab+\log _ac=\log _a(\mathit{bc})\)

\(\ln a-\ln b=\ln (\frac a b)\) \(\log _ab-\log _ac=\log _a(\frac b c)\)

\(\ln (a^b)=b\ln a\)      \(\log _a(b^c)=c\log _ab\)

\((\ln x)'=\frac 1 x\)       \((\log _ax)'=\frac 1{x\ln a}\)

\((\ln u(x))'=\frac{u'(x)}{u(x)}\) \((\log _au(x))'=\frac{u'(x)}{u(x)\ln a}\)

Astuce :

\[\lim \mathit{lorsque}x\mathit{tend}\mathit{vers}+\mathit{inf}(1+\frac k n)^n=e^k\]