Question

Tính

\(C=cos\dfrac{\pi}{5}+cos\dfrac{2\pi}{5}+cos\dfrac{3\pi}{5}+...+cos\dfrac{9\pi}{5}\)

Answer 1

\(...C=\left(cos\dfrac{\pi}{5}+cos\dfrac{9\pi}{5}\right)+\left(cos\dfrac{2\pi}{5}+cos\dfrac{8\pi}{5}\right)+\left(cos\dfrac{3\pi}{5}+cos\dfrac{7\pi}{5}\right)+\left(cos\dfrac{4\pi}{5}+cos\dfrac{6\pi}{5}\right)+cos\dfrac{5\pi}{5}\)

\(C=\left(cos\dfrac{\pi}{5}-cos\dfrac{\pi}{5}\right)+\left(cos\dfrac{2\pi}{5}-cos\dfrac{2\pi}{5}\right)+\left(cos\dfrac{3\pi}{5}-cos\dfrac{3\pi}{5}\right)+\left(cos\dfrac{4\pi}{5}-cos\dfrac{4\pi}{5}\right)+cos\pi=-1\)

Sử dụng công thức \(cos\left(\pi-x\right)=-cosx\) (2 góc bù nhau)

Answer 2

\(C=cos\left(\dfrac{\Omega}{5}\right)+cos\left(\dfrac{2\Omega}{5}\right)+...+cos\left(\dfrac{9\Omega}{5}\right)\)

\(=\left(cos\left(\dfrac{\Omega}{5}\right)+cos\left(\dfrac{9\Omega}{5}\right)\right)+\left(cos\left(\dfrac{2}{5}\Omega\right)+cos\left(\dfrac{8}{5}\Omega\right)\right)+\left(cos\left(\dfrac{3}{5}\Omega\right)+cos\left(\dfrac{7}{5}\Omega\right)\right)+cos\left(\dfrac{4}{5}\Omega\right)+cos\left(\dfrac{6}{5}\Omega\right)+cos\left(\dfrac{5}{5}\Omega\right)\)

\(=\dfrac{1}{2}\cdot cos\left(\dfrac{\left(\dfrac{9}{5}\Omega-\dfrac{\Omega}{5}\right)}{2}\right)\cdot cos\left(\dfrac{\left(\dfrac{9}{5}\Omega+\dfrac{\Omega}{5}\right)}{2}\right)+\dfrac{1}{2}\cdot cos\left(\dfrac{\left(\dfrac{8}{5}\Omega+\dfrac{2}{5}\Omega\right)}{2}\right)\cdot cos\left(\dfrac{\dfrac{8}{5}\Omega-\dfrac{2}{5}\Omega}{2}\right)+...+cos\left(\Omega\right)\)

\(=\dfrac{1}{2}\cdot cos\Omega\cdot cos\left(\dfrac{4}{5}\Omega\right)+\dfrac{1}{2}\cdot cos\Omega\cdot cos\left(\dfrac{3}{5}\Omega\right)+...+cos\Omega\)

\(=cos\Omega\left(\dfrac{1}{2}\cdot cos\left(\dfrac{4}{5}\Omega\right)+\dfrac{1}{2}\cdot cos\left(\dfrac{3}{5}\Omega\right)+\dfrac{1}{2}\cdot cos\left(\dfrac{2}{5}\Omega\right)+\dfrac{1}{2}\cdot cos\left(\dfrac{\Omega}{5}\right)\right)+\left(-1\right)\)

\(=\dfrac{-1}{2}\cdot\left[cos\left(\dfrac{4}{5}\Omega\right)+cos\left(\dfrac{\Omega}{5}\right)+cos\left(\dfrac{3}{5}\Omega\right)+cos\left(\dfrac{2}{5}\Omega\right)\right]-1\)

\(=\dfrac{-1}{4}\cdot\left[cos\left(\dfrac{\dfrac{4}{5}\Omega+\dfrac{\Omega}{5}}{2}\right)\cdot cos\left(\dfrac{\dfrac{4}{5}\Omega-\dfrac{\Omega}{5}}{2}\right)+cos\left(\dfrac{\dfrac{3}{5}\Omega+\dfrac{2}{5}\Omega}{2}\right)\cdot cos\left(\dfrac{\dfrac{3}{5}\Omega-\dfrac{2}{5}\Omega}{2}\right)\right]-1\)

\(=\dfrac{-1}{4}\cdot cos\left(\dfrac{\Omega}{2}\right)\left[cos\left(\dfrac{3}{10}\Omega\right)-cos\left(\dfrac{1}{10}\Omega\right)\right]-1\)

=-1