Update cheatsheet.fr.md

12.temporary_ins/69.waves/30.n3/20.overview/cheatsheet.fr.md

@@ -65,15 +65,20 @@ $`\begin{align} U(&x,t) = U_1(x,t) + U_2(x,t) \\
 &=A\;\big[\,cos(\underbrace{kx - \omega t}_{\text{  posons }\\ kx - \omega t \,=\, \alpha} + \varphi_1) + cos(\underbrace{kx - \omega t}_{=\; \alpha} + \varphi_1)\,\big]
 &\\
 &=A\;\big[\,cos\Big(\alpha + \dfrac{\varphi_1+\varphi_1}{2} + \dfrac{\varphi_2-\varphi_2}{2}\Big) \\
-&\quad\quad\quad + \,cos\Big(\alpha + \dfrac{\varphi_2+\varphi_2}{2} + \dfrac{\varphi_1-\varphi_1}{2}\Big)\,\Big]\\
+&\quad\quad\quad\quad + \,cos\Big(\alpha + \dfrac{\varphi_2+\varphi_2}{2} + \dfrac{\varphi_1-\varphi_1}{2}\Big)\,\Big]\\
 &\\
 &=A\;\big[\,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{=\;\alpha '} + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
-&\quad\quad\quad + \,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\text{nous avons posé }\\ \alpha + (\varphi_1+\varphi_2)/2\; = \;\alpha '} - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
+&\quad\quad\quad\quad + \,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\text{nous avons posé }\\ \alpha + (\varphi_1+\varphi_2)/2\; = \;\alpha '} - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
 &\\
 &=A\;\big[\,cos\Big(\alpha ' + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
-&\quad\quad\quad + \,cos\Big(\alpha ' - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
+&\quad\quad\quad\quad + \,cos\Big(\alpha ' - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
 &\\
-  \end{align}`$
+&=A\;\big[\,\underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1+\varphi_2}{2}\,-\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1+\varphi_2}{2}}_{\text{car }cos(a+b)=cos\,a\,cos\,b\;-\;sin\,a\,sin\,b}\big)\\
+&\quad\quad\quad\quad + \underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1+\varphi_2}{2}\,-\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1+\varphi_2}{2}}_{\text{car }cos(a-b)=cos\,a\,cos\,b\;+-\;sin\,a\,sin\,b}\big)\,\Big]\\
+&\\
+\end{align}`$
+
+
 * L'onde résultante $`U = U_1 + U_2`$ :   
   $`\begin{align} U(x,t) &= U_1(x,t) + U_2(x,t) \\
   &\\