Question

Determine que cos 3a = 4. cos ³ a - 3. cos a.

Answer 1

$\textbf{Dedu\c{c}\~ao pela identidade do cosseno da soma:}\\\\ \mathrm{\cos{(a+b)}=\cos{a}.\cos{b}-\sin{a}.\sin{b}}\\ \mathrm{\cos{3a}=\cos{(a+2a)}=\cos{a}.\cos{2a}-\sin{a}.\sin{2a}}\\\\ \mathrm{Lembrando\ que:}\\ \mathrm{*\ \cos{2a}=\cos^2{a}-\sin^2{a}=2\cos^2{a}-1}\\ \mathrm{*\ \sin{2a}=2\sin{a}\cos{a}}$

$\mathrm{\cos{3a}=\cos{a}.(2\cos^2{a}-1)-\sin{a}.2\sin{a}\cos{a}}\\ \mathrm{\cos{3a}=2\cos^3{a}-\cos{a}-2\cos{a}\sin^2{a}}\\ \mathrm{\cos{3a}=2\cos^3{a}-\cos{a}-2\cos{a}.(1-\cos^2{a})}\\ \mathrm{\cos{3a}=2\cos^3{a}-\cos{a}-2\cos{a}+2\cos^3{a}}\\ \mathbf{\cos{3a}=4\cos^3{a}-3\cos{a}}$

$\textbf{Dedu\c{c}\~ao pela f\'ormula de Euler:}\\\\ \mathrm{e^{i\phi}=\cos{\phi}+i\sin{\phi}\ \to\ \big(e^{i\phi}\big)^3=\cos{3\phi}+i\sin{3\phi}}\\ \mathrm{(\cos{\phi}+i\sin{\phi})^3=\cos{3\phi}+i\sin{3\phi}}\\ \mathrm{\cos^3{\phi}+3\cos^2{\phi}.i\sin{\phi}+3\cos{\phi}.(i\sin{\phi})^2+(i\sin{\phi})^3=\cos{3\phi}+i\sin{3\phi}}\\ \mathrm{\cos^3{\phi}+3\cos^2{\phi}.i\sin{\phi}-3\cos{\phi}.\sin^2{\phi}-i\sin{\phi}=\cos{3\phi}+i\sin{3\phi}}$
$\mathrm{(cos^3{\phi}-3\cos{\phi}.\sin^2{\phi})+i(3\cos^2{\phi}\sin{\phi}-\sin{\phi})=\cos{3\phi}+i\sin{3\phi}}\\ \mathrm{\Re{(z)}=\Re{(z)}\ \to\ \cos^3{\phi}-3\cos{\phi}.\sin^2{\phi}=\cos{3\phi}}\\ \mathrm{\cos^3{\phi}-3\cos{\phi}.(1-\cos^2{\phi})=\cos{3\phi}}\\ \mathrm{\cos^3{\phi}-3\cos{\phi}+3\cos^3{\phi}=\cos{3\phi}}\\ \mathbf{\cos{3\phi}=4\cos^3{\phi}-3\cos{\phi}}$