Update cheatsheet.fr.md

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

@@ -639,52 +639,10 @@ $`\quad\boldsymbol{\mathbf{=\color{brown}{2\,A\cdot cos\Big(\dfrac{\varphi_1-\va
   $`\quad \begin{align}=2\,A\;&c\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\\
      &\times \Big[ \,c\alpha\,c\Big(\dfrac{\varphi_1+\varphi_2}{2}\Big)-s\alpha\,s\Big(\dfrac{\varphi_1+\varphi_2}{2}\Big)\,\Big]
      \end{align}`$
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-]
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-\Big[\,&c\alpha\cdot 2\,cos\Big(\dfrac{\varphi_1+\varphi_2}{2}\Big)\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\\
-     &- s\alpha\cdot 2\,sin\Big(\dfrac{\varphi_1+\varphi_2}{2}\Big)\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\end{align}`$   
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-(c\alpha\;+\;i\,s\,\alpha) \cdot \big(\,(c\,\varphi_1\;+\;c\,\varphi_2)\;+\; i\,(s\,\varphi_1\;+\;s\,\varphi_2)\,\big)\big]`$
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-i\,s\,\varphi_1\;+\;c\,\varphi_2\;+\;i\,s\,\varphi_2)\,]`$ 
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-cos(\underbrace{(kx - \omega t}_{\color{blue}{=\; \alpha}} + \varphi_1)\,\big]
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-$`\begin{align} \quad &=A\;\big[\,e^{\,i\;(\underbrace{(kx - \omega t}_{\color{blue}{\text{  posons }\\ kx - \omega t \,=\, \alpha}}} + \varphi_1) + cos(\underbrace{(kx - \omega t}_{\color{blue}{=\; \alpha}} + \varphi_1)\,\big]
-&\\
-&=A\;\big[\,cos\Big(\alpha + \dfrac{\varphi_1+\varphi_1}{2} + \dfrac{\varphi_2-\varphi_2}{2}\Big) \\
-&\quad\quad\quad\quad + \,cos\Big(\alpha + \dfrac{\varphi_2+\varphi_2}{2} + \dfrac{\varphi_1-\varphi_1}{2}\Big)\,\Big]\\
-&\\
-&=A\;\big[\,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{=\;\alpha '}} + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
-&\quad\quad\quad\quad + \,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{\text{nous avons posé }\\ \alpha + (\varphi_1+\varphi_2)/2\; = \;\alpha '}} - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
-&\\
-&=A\;\big[\,cos\Big(\alpha ' + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
-&\quad\quad\quad\quad + \,cos\Big(\alpha ' - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
-&\\
-&=A\;\big[\,\underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,-\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a+b)\;=\;cos\,a\,cos\,b\;-\;sin\,a\,sin\,b}}\big)\\
-&\quad + \underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,+\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a-b)\;=\;cos\,a\,cos\,b\;+\;sin\,a\,sin\,b}}\big)\,\Big]\\
-&\\
-&=2\,A\cdot cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)
-\end{align}`$   
-<br>
-$`\quad\boldsymbol{\mathbf{=\color{brown}{2\,A\cdot cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big) \cdot cos\Big(}\color{blue}{\underbrace{\color{brown}{kx - \omega t + \dfrac{\varphi_1+\varphi_2}{2}}}_{\text{pulsation }\omega\text{ inchangée}}}\color{brown}{\Big)}}}`$
-
+  <br>
+  $`\quad =2\,A\;&c\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,&c\Big(\alpha + \dfrac{\varphi_1-\varphi_2}{2}\Big)`$
+  <br>
+  **$`\boldsymbol{\mathbf{\quad =2\,A\;&cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,&cos\Big(kx - \omega t + \dfrac{\varphi_1-\varphi_2}{2}\Big)}}`$**