Números Complexos

Sabendo que [latex]z=cis\theta [/latex] e que o valor da expressão [latex]sen^{2}(\theta) \cdot cos^{4}(\theta) + \frac{1}{32} \cdot cos(6\theta) + \frac{1}{16} \cdot cos(4\theta ) - \frac{1}{32} \cdot cos(2\theta )[/latex] possa ser escrito da forma [latex]\frac{a}{b}[/latex] onde [latex]a[/latex] e [latex]b[/latex] são números inteiros positivos primos entre si. Determine a soma dos algarismos do valor [latex]a^{2}+b^{2}[/latex].

Acho que dá para fazer por trigonometria, mas como o titulo é números complexos...




I) Tendo Z = cisθ,

[latex]\cos \theta = \frac{Z^{1} + Z^{-1}}{2}[/latex]

[latex]\sin \theta = \frac{Z^{1} - Z^{-1}}{2i}[/latex]

[latex]\cos 2 \theta = \frac{Z^{2} + Z^{-2}}{2}[/latex]

[latex]\cos 4 \theta = \frac{Z^{4} + Z^{-4}}{2}[/latex]

[latex]\cos 6 \theta = \frac{Z^{6} + Z^{-6}}{2}[/latex]




II) Com isso,

[latex]K = \sin^{2} (\theta )\cdot \cos^{4} (\theta ) + \frac{1}{32}\cdot \cos (6\theta ) + \frac{1}{16}\cdot \cos (4\theta ) - \frac{1}{32}\cdot \cos (2\theta )[/latex]

[latex]K =\left ( \frac{Z^{1} - Z^{-1}}{2i} \right )^{2}\cdot \left ( \frac{Z^{1} + Z^{-1}}{2} \right )^{4} + \frac{1}{32}\cdot \frac{Z^{6} + Z^{-6}}{2} + \frac{1}{16}\cdot \frac{Z^{4} + Z^{-4}}{2} - \frac{1}{32}\cdot \frac{Z^{2} + Z^{-2}}{2}[/latex]

[latex]K =\left ( \frac{Z^{1} - Z^{-1}}{2i} \right )^{2}\cdot \left ( \frac{Z^{1} + Z^{-1}}{2} \right )^{4} + \frac{Z^{6} + 2Z^{4} - Z^{2} + Z^{-6} + 2Z^{-4} - Z^{-2}}{64}[/latex]

[latex]K = -\frac{Z^{6} + 2Z^{4} - Z^{2} + Z^{-6} + 2Z^{-4} - Z^{-2} - 4}{64} + \frac{Z^{6} + 2Z^{4} - Z^{2} + Z^{-6} + 2Z^{-4} - Z^{-2}}{64}[/latex]

[latex]K = \frac{4}{64} = \frac{1}{16}[/latex]

Portanto,

[latex]\begin{matrix} a = 1\\ b = 16 \end{matrix}[/latex]




III) Logo,

[latex]S = 1^2 + 16^2 = 1 + 256 = 257[/latex]


[latex]R = 2 + 5 + 7 = 14[/latex]

Por trigonometria...

[latex]K = \sin^{2} (\theta )\cdot \cos^{4} (\theta ) + \frac{1}{32}\cdot \cos (6\theta ) + \frac{1}{16}\cdot \cos (4\theta ) - \frac{1}{32}\cdot \cos (2\theta )[/latex]


[latex]K = \frac{\left (2\sin (\theta ) \cos (\theta ) \right )^{2}\cdot \cos^{2} (\theta )}{4} + \frac{\cos (6\theta )  + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{\sin^{2} (2\theta ) \cdot \cos^{2} (\theta )}{4} + \frac{\cos (6\theta )  + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{\frac{1 - \cos (4\theta )}{2} \cdot \frac{1 + \cos (2\theta )}{2}}{4} + \frac{\cos (6\theta ) + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{ 1 - \cos (4\theta ) + \cos (2\theta ) - \cos (4\theta ) \cdot \cos (2\theta )}{16} + \frac{\cos (6\theta )  + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{1}{16} + \frac{ - 2\cos (4\theta ) + 2\cos (2\theta ) - 2\cos (4\theta ) \cdot \cos (2\theta )}{32} + \frac{\cos (6\theta ) + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{1}{16} + \frac{ - 2\cos (4\theta ) + 2\cos (2\theta ) - \cos (2\theta ) - \cos (6\theta )}{32} + \frac{\cos (6\theta ) + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{1}{16} + \frac{\cos (2\theta ) - 2\cos (4\theta ) - \cos (6\theta )}{32} + \frac{\cos (6\theta ) + 2\cos (4\theta ) - \cos (2\theta )}{32}[/latex]


[latex]K = \frac{1}{16}[/latex]

Muito obrigado pelas duas resoluções!