Correction exercice I

En utilisant les formules de Moivre, linéarisons les expressions suivantes :
a) \(A(x) = {\cos ^3}x\)
\(A(x) = \) \({\left( {\frac{{Z + \overline Z }}{2}} \right)^3}\) \( = \frac{1}{8}({Z^3} + \) \(3{Z^2}\overline Z + \) \(3Z{\overline Z ^2} + \) \({\overline Z ^3})\)
\({Z^n} + {\overline Z ^n}\) \( = 2\cos \left( {n\theta } \right)\)
\({Z^3} + {\overline Z ^3}\) \( = 2\cos \left( {3\theta } \right)\)
\({Z^3} + {\overline Z ^3}\) \( = 2\cos \left( {3\theta } \right)\)
\(Z + \overline Z = \) \(2\cos \left( \theta \right)\)
\(A(\theta) = \) \(\frac{1}{8}\left( {{Z^3} + {{\overline Z }^3}} \right) + \) \(\frac{3}{8}Z\overline Z \left( {Z + \overline Z } \right)\)
\(A(\theta) = \) \(\frac{1}{8}2\cos \left( {3\theta } \right)\) \( + \frac{3}{8}2\cos \left( \theta \right)\) \( = \frac{1}{4}\cos \left( {3\theta } \right)\) \( + \frac{3}{4}\cos \left( \theta \right)\)

\(A(x) = \) \(\frac{1}{4}\cos \left( {3x} \right)\) \( + \frac{3}{4}\cos \left( x \right)\)

b) \(B(x) = {\sin ^3}x\)
\(B(x) = \) \({\sin ^3}x = \) \({\left( {\frac{{Z + \overline Z }}{2}} \right)^3}\)
\({Z^n} - {\overline Z ^n}\) \( = 2i\sin \left( {n\theta } \right)\)
\({Z^n} \times {\overline Z ^n} = 1\)
\(B(x) = \) \({\left( {\frac{{Z - \overline Z }}{2}} \right)^3}\) \( = \frac{1}{{{{\left( {2i} \right)}^3}}}\) \((2({Z^3} - {\overline Z ^3})\) \( + 3(Z - \overline Z ))\)
\(B(x) = \) \(\frac{1}{{ - 8i}}(2i\sin 3x)\) \( + \frac{1}{{ - 8i}}(6i\sin x)\)

\(B(x) = \) \( - \frac{1}{4}\sin 3x\) \( - \frac{3}{4}\sin x\)

c) \(C(x) = {\cos ^5}\frac{x}{2}\)
\(C(x) = \) \(\frac{1}{{32}}\left( {{Z^5} + {{\overline Z }^5}} \right) + \) \(\frac{1}{{32}}5Z\overline Z \left( {{Z^3} + {{\overline Z }^3}} \right)\) \( + \frac{1}{{32}}10{Z^2}{\overline Z ^2}\) \(\left( {Z + \overline Z } \right)\)
\(C(x) = \) \(\frac{1}{{32}}2\cos 5\frac{x}{2}\) \( + \frac{1}{{32}}10\cos 3\frac{x}{2}\) \( + \frac{1}{{32}}20\cos \frac{x}{2}\)

\(C(x) = \) \(\frac{1}{{16}}\cos \frac{{5x}}{2} + \) \(\frac{5}{{16}}\cos \frac{{3x}}{2} + \) \(\frac{{10}}{{16}}\cos \frac{x}{2}\)

d) \(D(x) = \) \({\cos ^3}x{\sin ^3}x\)
\(D(x) = \) \({\left( {\frac{{Z + \overline Z }}{2}} \right)^3}\) \({\left( {\frac{{Z - \overline Z }}{{2i}}} \right)^3}\) \( = \frac{1}{{ - 64i}}\left( {{Z^6} - {{\overline Z }^6}} \right)\) \( - \frac{3}{{ - 64i}}{Z^2}{\overline Z ^2}\) \(\left( {{Z^2} - {{\overline Z }^2}} \right)\)
\(D(x) = \) \(\frac{1}{{ - 64i}}\left( {2i\sin 6x} \right)\) \( - \frac{3}{{ - 64i}}\left( {2i\sin 2x} \right)\)

\(D(x) = \) \( - \frac{1}{{32}}\sin 6x + \) \(\frac{3}{{32}}\sin 2x\)

Exercice II

En utilisant la formule d'Euler, linéarisons les expressions suivantes :
a) \(A(x) = {\cos ^3}x\)
\(A(x) = \) \(\frac{1}{8}{\left( {{e^{ix}} + {e^{ - ix}}} \right)^3}\) \(\frac{1}{8}\left( {{e^{3ix}} + {e^{ - 3ix}}} \right)\) \( + \frac{3}{8}\left( {{e^{ix}} + {e^{ - ix}}} \right)\)

• \({e^{nix}} + {e^{ - nix}}\) \( = 2\cos nx\)
• \({e^{nix}} \times {e^{ - nix}}\) \( = 1\)

\(A(x) = \) \(\frac{1}{4}\left( {\cos 3x} \right)\) \( + \frac{3}{4}\left( {\cos x} \right)\)

b) \(B(x) = {\sin ^3}x\)
\(B(x) = \) \(\sin 3x = \) \({\left( {\frac{{{e^{ix}} - {e^{ - ix}}}}{{2i}}} \right)^3}\) \( = \frac{1}{{ - 8i}}\) \( = \frac{1}{{ - 8i}}\) \( + \frac{1}{{ - 8i}}\) \(\left( {{e^{ix}} - {e^{ - ix}}} \right)\)

\(B(x) = - \) \(\frac{1}{4}\sin 3x - \) \(\frac{3}{4}\sin x\)

c) \(C(x) = {\cos ^5}\frac{x}{2}\)
d) \(D(x) = \) \({\cos ^3}x{\sin ^3}x\)

Correction exercice III

Linéarisons en utilisant :
a) Les formules d’Euler;
\({\cos ^2}x = \) \({\left( {\frac{{{e^{ix}} + {e^{ - ix}}}}{2}} \right)^2}\) \( = \frac{1}{4}({\left( {{e^{ix}}} \right)^2}\) \( + 2{e^{ix}}{e^{ - ix}}\) \( + {\left( {{e^{ - ix}}} \right)^2} = \) \(\frac{1}{4}\left( {{e^{i2x}} + {e^{ - 2ix}}} \right)\) \( + \frac{1}{2}\)
b) Les formules usuelles trigonométriques
\(\cos (x + x) = \) \(\cos (2x) = \) \({\cos ^2}x - {\sin ^2}x\) \( = {\cos ^2}x - \) \((1 - {\cos ^2}x)\) \( = 2{\cos ^2}x - 1\)
\({\cos ^2}x = \) \(\frac{1}{2}\cos 2x + \frac{1}{2}\)

Linéarisons en utilisant :
a) Les formules d’Euler;
\({\sin ^3}x\)
\({\sin ^3}x = \) \({\left( {\frac{{{e^{ix}} - {e^{ - ix}}}}{{2i}}} \right)^3}\) \( = - \frac{1}{4}\) \(\left( {\frac{{{e^{i3x}} - {e^{ - i3x}}}}{{2i}}} \right)\) \( - \frac{3}{4}\) \(\left( {\frac{{{e^{ix}} - {e^{ - ix}}}}{{2i}}} \right)\)
\({\sin ^3}x = \) \( - \frac{1}{4}\sin 3x\) \( - \frac{1}{4}\sin 3x\)

Correction exercice IV

Linéarisons \(2\left( {1 + {{\sin }^2}x} \right)\) \({\cos ^2}x\)
\(2\left( {1 + {{\sin }^2}x} \right)\) \({\cos ^2}x\) \( = 2\) \(\left( {1 + {{\left( {\frac{{{e^{ix}} - {e^{ - ix}}}}{{2i}}} \right)}^2}} \right)\) \({\left( {\frac{{{e^{ix}} + {e^{ - ix}}}}{{2i}}} \right)^2}\) \( = \frac{1}{8}\) \(\left( {6 - {e^{2ix}} - {e^{ - 2ix}}} \right)\) \(\left( {2 + {e^{2ix}} + {e^{ - 2ix}}} \right)\)
\(2\left( {1 + {{\sin }^2}x} \right)\) \({\cos ^2}x = \) \(\frac{1}{4}(5 + 4\cos 2x\) \( - \cos 4x)\)