Update cheatsheet.fr.md

12.temporary_ins/07.geometry-coordinates-prop2/40.n4/20.overview/cheatsheet.fr.md

@@ -75,77 +75,102 @@ $`v_2=v\cdot\sin(a + \theta + \alpha)`$
 $`v^2=v\cdot\dfrac{\sin(\theta + \alpha)}{\cos(a)}`$
 
 
-$`u_1\,v^1\,+\, u_2\,v^2\,=u\cdot\cos(\alpha)\cdot v\cdot\dfrac{\cos(a + \theta + \alpha)}
-{\cos(a)} + u\cdot\sin(a + \alpha) \cdot v\cdot\dfrac{\sin(\theta + \alpha)}{\cos(a)}`$
+$`\begin{equation}\begin{split}
+u_1v^1+u_2v^2\,&=u\cdot\cos(\alpha)\cdot v\cdot\dfrac{\cos(a + \theta + \alpha)}{\cos(a)} \\
+&\quad + u\cdot\sin(a + \alpha) \cdot v\cdot\dfrac{\sin(\theta + \alpha)}{\cos(a)}
+\end{split}\end{equation}`$
 
-$`\quad= u\,v\,\dfrac{1}{\cos(a)} \,\left[\cos(\alpha)\,\cos(a + \theta + \alpha)\,+\,\sin(a + \alpha)\,\sin(\theta + \alpha)\right]`$
+$`\begin{equation}\begin{split}
+u_1v^1+u_2v^2\,&= u\,v\,\dfrac{1}{\cos(a)} \times \\
+& \quad \left[+\cos(\alpha)\,\cos(a + \theta + \alpha) \right. \\
+& \left.\quad\;+\,\sin(a + \alpha)\,\sin(\theta + \alpha)\,\right]
+\end{split}\end{equation}`$
 
-On a
+En utilisant les relations trigonométriques
 
-$`\cos(a + \theta + \alpha)=\cos\left[a + (\theta + \alpha)\right]=\cos(a)\,\cos(\theta + \alpha)-\sin(a)\,\sin(\theta + \alpha)`$
+$`\cos(\theta + \alpha)=\cos(\theta)\cos(\alpha)-\sin(\theta)\sin(\alpha)`$
 
 $`\sin(a + \alpha)=\sin(a)\cos(\alpha)+\sin(\alpha)\cos(a)`$
 
 $`\sin(\theta + \alpha)=\sin(\theta)\cos(\alpha)+\sin(\alpha)\cos(\theta)`$
 
-donc
-
-$`u_1\,v^1\,+\, u_2\,v^2\,= u\,v\,\dfrac{1}{\cos(a)} \,\left[\cos(a)\cos(\alpha)\cos(\theta + \alpha)\,-\,\sin(a)\sin(\alpha)\sin(\theta + \alpha)
-\,+\,\sin(a)\,\cos^2(\alpha)\,\sin(\theta)
-\,+\,\sin(a)\,\cos(\alpha)\,\sin(\alpha)\,\cos(\theta)
-\,+\,\cos(a)\,\sin(\alpha)\,\cos(\alpha)\,\sin(\theta)
-\,+\,\cos(a)\,\sin^2(\alpha)\,\cos(\theta)\right]`$
-
-De plus
-
-$`\cos(\theta + \alpha)=\cos(\theta)\,\cos(\alpha)\,-\,\sin(\theta)\,\sin(\alpha)`$
-
-$`\sin(\theta + \alpha)=\sin(\theta)\,\cos(\alpha)\,+\,\sin(\alpha)\,\cos(\theta)`$
-
-donc
-
-$`u_1\,v^1\,+\, u_2\,v^2\,= u\,v\,\dfrac{1}{\cos(a)} \,\left[\cos(a)\cos(\alpha)\cos(\theta)\,\cos(\alpha)
-\,-\,\cos(a)\cos(\alpha)\sin(\theta)\,\sin(\alpha)
-\,-\,\sin(a)\sin(\alpha)\sin(\theta)\,\cos(\alpha)
-\,-\,\sin(a)\sin(\alpha)\sin(\alpha)\,\cos(\theta)
-\,+\,\sin(a)\,\cos^2(\alpha)\,\sin(\theta)
-\,+\,\sin(a)\,\cos(\alpha)\,\sin(\alpha)\,\cos(\theta)
-\,+\,\cos(a)\,\sin(\alpha)\,\cos(\alpha)\,\sin(\theta)
-\,+\,\cos(a)\,\sin^2(\alpha)\,\cos(\theta)\right]`$
-
-$`u_1\,v^1\,+\, u_2\,v^2\,= u\,v\,\dfrac{1}{\cos(a)} \,
-\left[\cos(a)\cos^2(\alpha)\cos(\theta)
-\,-\,\cos(a)\cos(\alpha)\sin(\theta)\,\sin(\alpha)
-\,-\,\sin(a)\sin(\alpha)\sin(\theta)\,\cos(\alpha)
-\,-\,\sin(a)\sin^2(\alpha)\,\cos(\theta)
-\,+\,\sin(a)\,\cos^2(\alpha)\,\sin(\theta)
-\,+\,\sin(a)\,\cos(\alpha)\,\sin(\alpha)\,\cos(\theta)
-\,+\,\cos(a)\,\sin(\alpha)\,\cos(\alpha)\,\sin(\theta)
-\,+\,\cos(a)\,\sin^2(\alpha)\,\cos(\theta)\right]`$
-
-à vérifier et terminer
-
-$`u_1\,v^1\,+\, u_2\,v^2\,= u\,v\,\dfrac{1}{\cos(a)} \,\left[\cos(a)\cos^2(\alpha)\cos(\theta)
-\,-\,\cancel{\cos(a)\cos(\alpha)\sin(\theta)\,\sin(\alpha)}
-\,-\,\sin(a)\sin(\alpha)\sin(\theta)\,\cos(\alpha)
-\,-\,\sin(a)\sin^2(\alpha)\,\cos(\theta)
-\,+\,\xcancel{\sin(a)\,\cos^2(\alpha)\,\sin(\theta)}
-\,+\,\sin(a)\,\cos(\alpha)\,\sin(\alpha)\,\cos(\theta)
-\,+\,\cancel{\cos(a)\,\sin(\alpha)\,\cos(\alpha)\,\sin(\theta)}
-\,+\,\cos(a)\,\sin^2(\alpha)\,\cos(\theta)\right]`$
-
-
-$`\begin{multline}
-u_1\,v^1\,+\, u_2\,v^2\,= u\,v\,\dfrac{1}{\cos(a)} \,\left[\cos(a)\cos^2(\alpha)\cos(\theta)\\
-\,-\,\cancel{\cos(a)\cos(\alpha)\sin(\theta)\,\sin(\alpha)}\\
-\,-\,\sin(a)\sin(\alpha)\sin(\theta)\,\cos(\alpha)\\
-\,-\,\sin(a)\sin^2(\alpha)\,\cos(\theta)\\
-\,+\,\xcancel{\sin(a)\,\cos^2(\alpha)\,\sin(\theta)}\\
-\,+\,\sin(a)\,\cos(\alpha)\,\sin(\alpha)\,\cos(\theta)\\
-\,+\,\cancel{\cos(a)\,\sin(\alpha)\,\cos(\alpha)\,\sin(\theta)}\\
-\,+\,\cos(a)\,\sin^2(\alpha)\,\cos(\theta)\right]\\
-\end{multline}`$
-
+qui impliquent
+
+$`\begin{equation}\begin{split}
+\cos(a\,&\,+ \theta + \alpha)=\cos\left[a + (\theta + \alpha)\right] \\
+& \\
+&=\cos(a)\,\cos(\theta + \alpha)-\sin(a)\,\sin(\theta + \alpha) \\
+& \\
+&=+\cos(a)\,[\cos(\theta)\cos(\alpha)-\sin(\theta)\sin(\alpha)] \\
+& \quad-\sin(a)\,[\sin(\theta)\cos(\alpha)+\sin(\alpha)\cos(\theta)]\\
+& \\
+&=+\cos(a)\cos(\theta)\cos(\alpha) \\
+&\quad-\cos(a)\sin(\theta)\sin(\alpha) \\
+& \quad-\sin(a)\sin(\theta)\cos(\alpha)\\
+& \quad-\sin(a)\sin(\alpha)\cos(\theta)]
+\end{split}\end{equation}`$
+
+et
+
+$`\begin{equation}\begin{split}
+\sin(a +& \alpha)\,\sin(\theta + \alpha) \\
+&=[\sin(a)\cos(\alpha)+\sin(\alpha)\cos(a)] \\
+& \quad\times[\sin(\theta)\cos(\alpha)+\sin(\alpha)\cos(\theta)] \\
+& \\
+&= + \sin(a)\cos(\alpha)\sin(\theta)\cos(\alpha) \\
+& \quad + \sin(a)\cos(\alpha)\sin(\alpha)\cos(\theta) \\
+& \quad + \sin(\alpha)\cos(a)\sin(\theta)\cos(\alpha) \\
+& \quad +\sin(\alpha)\cos(a)\sin(\alpha)\cos(\theta)
+\end{split}\end{equation}`$
+
+nous obtenons
+
+$`\begin{equation}
+\begin{split}
+u_1v^1+u_2v^2 &= u\,v\,\dfrac{1}{\cos(a)} \times \\
+& \left[\,+\,\cos(\alpha)\cos(a)\cos(\theta)cos(\alpha)\right.\\
+&\;-\,\cos(\alpha)\cos(a)\sin(\theta)\sin(\alpha)\\
+&\;-\,\cos(\alpha)\sin(a)\sin(\theta)\cos(\alpha)\\
+&\;-\,\cos(\alpha)\sin(a)\sin(\alpha)\cos(\theta)\\
+&\;+\, \sin(a)\cos(\alpha)\sin(\theta)\cos(\alpha)\\
+&\;+\,\sin(a)\cos(\alpha)\sin(\alpha)\cos(\theta)\\
+&\;+\,\sin(\alpha)\cos(a)\sin(\theta)\cos(\alpha)\\
+ & \left.\;\;+\,\sin(\alpha)\cos(a)\sin(\alpha)\cos(\theta)\,\right]
+\end{split}
+\end{equation}`$
+
+$`\begin{equation}
+\begin{split}\require{cancel}
+u_1v^1+u_2v^2 &= u\,v\,\dfrac{1}{\cos(a)} \times \\
+& \left[\,+\,\cos(\alpha)\cos(a)\cos(\theta)cos(\alpha)\right.\\
+&\;-\,\cos(\alpha)\cancel{\cos(a)\sin(\theta)}\sin(\alpha)\\
+&\;-\,\cos(\alpha)\cancel{\sin(a)\sin(\theta)}\cos(\alpha)\\
+&\;-\,\cos(\alpha)\cancel{\sin(a)\sin(\alpha)}\cos(\theta)\\
+&\;+\,\sin(a)\cancel{\cos(\alpha)\sin(\theta)}\cos(\alpha)\\
+&\;+\,\sin(a)\cancel{\cos(\alpha)\sin(\alpha)}\cos(\theta)\\
+&\;+\,\sin(\alpha)\cancel{\cos(a)\sin(\theta)}\cos(\alpha)\\
+ & \left.\;\;+\,\sin(\alpha)\cos(a)\sin(\alpha)\cos(\theta)\,\right]
+\end{split}
+\end{equation}`$
+
+$`\begin{equation}
+\begin{split}\require{cancel}
+u_1v^1+u_2v^2 &= u\,v\,\dfrac{1}{\cos(a)} \times \\
+& \quad\left[\,+\,\cos^2(\alpha)\cos(a)\cos(\theta)\right.\\
+& \quad\left.\;\;+\,\sin^2(\alpha)\cos(a)\cos(\theta)\,\right] 
+\end{split}\end{equation}`$
+
+$`\begin{equation}
+\begin{split}\require{cancel}
+u_1v^1+u_2v^2 &= u\,v\,\dfrac{1}{\cos(a)} \times \\
+& \left[\left(\cos^2(\alpha) + \sin^2(\alpha)\right)\,\cos(a)\cos(\theta)\right]
+\end{split}\end{equation}`$
+
+$`u_1v^1+u_2v^2 = u\,v\,\dfrac{\cos(a)\cos(\theta)}{\cos(a)}`$
+
+$`u_1v^1+u_2v^2 = u\,v\,\cos(\theta)`$
+
+Les choses se mettent en place pour tout ce niveau 4... je commence à voir une organisation..