Question

(cosx+isenx)³=cos3x+isen3x

Alguém consegue resolver essa?

Answer 1

Provar que $\mathsf{(cos\,x+i\,sen\,x)^{3}=cos\,3x+i\,sen\,3x}$ equivale a provar que

$\bullet\,\,\mathsf{cos\,3x=cos^{3}x-3cosx\,sen^{2}x}\\\\\bullet\,\,\mathsf{sen\,3x=-sen^{3}x+3cos^{2}x\,sen\,x }$

pois sabemos que $\mathsf{(a+b)^{3}=a^{3}+b^{3}+3a^{2}b+3ab^{2}}$, então

$\mathsf{(cos\,x+i\,sen\,x)^{3}=cos^{3}x+(i\,sen\,x)^{3}+3cos^{2}x\,i\,sen\,x+3cos\,x\,(i\,\,sen\,x)^{2}}\\\\\mathsf{(cos\,x+i\,sen\,x)^{3}=cos^{3}x+i^{3}sen^{3}x+3i\,cos^{2}x\,sen\,x+3cos\,x\,(i^{2})\,sen^{2}x}\\\\\mathsf{(cos\,x+i\,sen\,x)^{3}=cos^{3}x-i\,sen^{3}x+3i\,cos^{2}x\,sen\,x-3\,cos\,x\,sen^{2}x}$

Agrupando partes real e imaginária:

$\mathsf{(cos\,x+i\,sen\,x)^{3}=\big(cos^{3}x-3\,cos\,x\,sen^{2}x)+i\big(-sen^{3}x+3\,cos^{2}x\,sen\,x\big)}$

Como dois números complexos são iguais se, e somente se, suas partes real e imaginária são iguais, temos que

$\mathsf{(cos\,3x+i\,sen\,3x)^{3}=cos\,3x+i\,sen\,3x}$

se e somente se

$\mathsf{cos\,3x=cos^{3}x-3\,cos\,x\,sen^{2}x\,\,\,e\,\,\,sen\,3x=-sen^{3}x+3\,cos^{2}x\,sen\,x}$
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Sabemos que

$\bullet\,\,\mathsf{sen\,(a+b)=sen\,a\,cos\,b+sen\,b\,cos\,a}\\\\\bullet\,\,\mathsf{cos\,(a+b)=cos\,a\,cos\,b+sen\,a\,sen\,b}$

Daí, tiramos que

$\bullet\,\,\mathsf{sen\,2x=sen\,(x+x)=sen\,x\,cos\,x+sen\,x\,cos\,x=2\,sen\,x\,cos\,x}\\\\\bullet\,\,\mathsf{cos\,2x=cos\,(x+x)=cos\,x\,cos\,x-sen\,x\,sen\,x=cos^{2}x-sen\,^{2}x}$

Com isso, vamos calcular $\mathsf{cos\,3x}$

$\mathsf{cos\,3x=cos\,(2x+x)}\\\\\mathsf{cos\,3x=cos\,2x\,cos\,x-sen\,2x\,sen\,x}\\\\\mathsf{cos\,3x=\big(cos^{2}x-sen^{2}x\big)cos\,x-(2\,sen\,x\,cos\,x)sen\,x}\\\\\mathsf{cos\,3x=cos^{3}x-sen^{2}x\,cos\,x-2sen^{2}x\,cos\,x}\\\\\\\boxed{\boxed{\mathsf{cos\,3x=cos^{3}x-3\,sen^{2}x\,cos\,x}}}$

Agora calcularemos $\mathsf{sen\,3x}$:

$\mathsf{sen\,3x=sen\,(2x+x)}\\\\\mathsf{sen\,3x=sen\,2x\,cos\,x+sen\,x\,cos\,2x}\\\\\mathsf{sen\,3x=(2\,sen\,x\,cos\,x)cos\,x+sen\,x\big(cos^{2}x-sen^{2}x)}\\\\\mathsf{\mathsf{sen\,3x=2\,sen\,x\,cos^{2}x+sen\,x\,cos^{2}x-sen^{3}x}}\\\\\\\boxed{\boxed{\mathsf{sen\,3x=-sen^{3}x+3\,sen\,x\,cos^{2}x}}}$