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por Aeron945 19/5/2016, 12:40 am

Calcule sen⁴(pi/16) + sen⁴(3pi/16) + sen⁴(5pi/16) + sen⁴(7pi/16)

por **Carlos Adir** 20/5/2016, 10:16 pm

$\\ \mathrm{sen \frac{\pi}{16}=cos\frac{7\pi}{16}} \\ \mathrm{e^{ix}=cos x + i \cdot sen x} \\ \mathrm{\Rightarrow cos \frac{7\pi}{16}=\dfrac{1}{2}\left(e^{i\cdot \frac{7\pi}{16}}+e^{-i \cdot \frac{7\pi}{16}} \right )} \\ \mathrm{\Rightarrow cos^2\frac{7\pi}{16}=\dfrac{1}{4}\left(e^{i \cdot \frac{7\pi}{8}}+2+e^{-i \cdot \frac{7\pi}{8}} \right )} \\ \mathrm{\Rightarrow cos^4 \frac{7\pi}{{16}}=\dfrac{1}{16}\left(e^{i \cdot \frac{7\pi}{4}} +4e^{i \cdot \frac{7\pi}{8}}+6 +4e^{-i \cdot \frac{7\pi}{8}}+e^{-i \cdot \frac{7\pi}{4}}\right )} \\ \mathrm{cos^4 \frac{7\pi}{{16}}=\dfrac{3}{8}+\dfrac{1}{8}\left(cos\frac{7\pi}{4}+4cos\frac{7\pi}{8} \right )}$
De modo análogo com os demais, temos:
$\\ \mathrm{sen \frac{3\pi}{16}=cos\frac{5\pi}{16}} \\ \mathrm{cos^4 \frac{5\pi}{{16}}=\dfrac{3}{8}+\dfrac{1}{8}\left(cos\frac{5\pi}{4}+4cos\frac{5\pi}{8} \right )} \\ \mathrm{sen \frac{5\pi}{16}=cos\frac{3\pi}{16}} \\ \mathrm{cos^4 \frac{3\pi}{{16}}=\dfrac{3}{8}+\dfrac{1}{8}\left(cos\frac{3\pi}{4}+4cos\frac{3\pi}{8} \right )} \\ \mathrm{sen \frac{7\pi}{16}=cos\frac{\pi}{16}} \\ \mathrm{cos^4 \frac{\pi}{{16}}=\dfrac{3}{8}+\dfrac{1}{8}\left(cos\frac{\pi}{4}+4cos\frac{\pi}{8} \right )}$

Logo, a soma fica:
$\\ S = 4 \cdot \dfrac{3}{8}+\dfrac{1}{8}\left(cos \frac{\pi}{4}+cos\frac{3\pi}{4}+cos\frac{5\pi}{4}+cos \frac{7\pi}{{4}} \right )+\dfrac{1}{2}\left(cos \frac{\pi}{8}+cos\frac{3\pi}{8} +cos\frac{5\pi}{8}+cos\frac{7\pi}{8}\right ) \\ \boxed{S = \frac{3}{2}}$
Isso porque:
$\\ cos \frac{\pi}{8}+cos \frac{7\pi}{8}=0 \\ cos \frac{3\pi}{8}+cos \frac{5\pi}{8}=0 \\ cos \frac{\pi}{4}+cos \frac{3\pi}{4}=0 \\ cos \frac{5\pi}{4}+cos \frac{7\pi}{4} = 0$

logo, a resposta é 3/2.

por Aeron945 21/5/2016, 6:43 pm

Muito obrigado!!!

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