Question

If the sum of the roots of the equation $\cos 4x+6=7\cos 2x$ in the interval $[0,314]$ is $k\pi$, $k\in R$. Find $k$.

Answer 1

$\pi =3.14\Rightarrow 314=100\pi \Rightarrow x\in [0,100\pi ]$

$\cos 4x+6=7\cos 2x2{\cos }^{2}2x-1+6=7\cos 2x2{\cos }^{2}2x-7\cos 2x+5=02{\cos }^{2}2x-2\cos 2x-5\cos 2x+5=02\cos 2x(\cos 2x-1)-5(\cos 2x-1)=0(2\cos 2x-5)(\cos 2x-1)=0\Rightarrow \cos 2x-1=0\cos 2x=1x=0,4\pi ,8\pi ,12\pi ...........100\pi S=0+4\pi +8\pi +.........100\pi S=\pi (4+8+\mathrm{12........100})S=\pi \frac{25}{2}(4+100)S=1300\pi$

$k=1300$