Update cheatsheet.fr.md

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

@@ -67,14 +67,14 @@ $`\begin{align} \quad &=A\;\big[\,cos(\underbrace{kx - \omega t}_{\color{blue}{\
 &=A\;\big[\,cos\Big(\alpha + \dfrac{\varphi_1+\varphi_1}{2} + \dfrac{\varphi_2-\varphi_2}{2}\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\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(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{=\;\alpha '}} + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
+&\quad\quad\quad\quad + \,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{\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\quad + \,cos\Big(\alpha ' - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
 &\\
-&=A\;\big[\,\underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,-\,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 + \underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,+\,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]\\
+&=A\;\big[\,\underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,-\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a+b)\;=\;cos\,a\,cos\,b\;-\;sin\,a\,sin\,b}}\big)\\
+&\quad + \underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,+\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a-b)\;=\;cos\,a\,cos\,b\;+\;sin\,a\,sin\,b}}\big)\,\Big]\\
 &\\
 &=2\,A\cdot cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)
 \end{align}`$