How do you solve cos 2x(2cosx+1) = 0 in the interval 0 to 2pi?

Feb 29, 2016

$\setminus \cos \left(2 x\right) \left(2 \setminus \cos \left(x\right) + 1\right) = 0 , 0 \setminus \le \setminus x \setminus \le \setminus 2 \setminus \pi$ =

x=\frac{2\pi } {3},x=\frac{\pi }{4},x=\frac{3\pi }{4},x=\frac{5\pi }{4},x=\frac{4\pi }{3},x=\frac{7\pi }{4}

Explanation:

Solving each part separately,
$\setminus \cos \left(2 x\right) = 0 \mathmr{and} 2 \setminus \cos \left(x\right) + 1 = 0$

Now,
$\setminus \cos \left(2 x\right) = 0 , 0 \setminus \le x \setminus \le 2 \setminus \pi$

General solutions for $\cos \left(2 x\right) = 0$,
$\cos \left(2 x\right) = 0$ : $2 x = \frac{\pi}{2} + 2 \pi n$ , $2 x = \frac{3 \pi}{2} + 2 \pi n$

Solving we get,
$2 x = \setminus \frac{\setminus \pi}{2} + 2 \setminus \pi n : x = \setminus \frac{4 \setminus \pi n + \setminus \pi}{4}$

$2 x = \setminus \frac{3 \setminus \pi}{2} + 2 \setminus \pi n : x = \setminus \frac{4 \setminus \pi n + 3 \setminus \pi}{4}$

$x = \setminus \frac{4 \setminus \pi n + \setminus \pi}{4} , x = \setminus \frac{4 \setminus \pi n + 3 \setminus \pi}{4}$

So,solution for the range $0 \setminus \le x \setminus \le 2 \setminus \pi$

$x = \setminus \frac{\setminus \pi}{4} , x = \setminus \frac{3 \setminus \pi}{4} , x = \setminus \frac{5 \setminus \pi}{4} , x = \setminus \frac{7 \setminus \pi}{4}$

Again,we have,
$2 \setminus \cos \left(x\right) + 1 = 0 , 0 \setminus \le x \setminus \le 2 \setminus \pi$

Isolating $\cos \left(x\right)$
$\cos \left(x\right) = - \frac{1}{2}$

General solutions for $\cos \left(x\right)$ = -1/2#

$\setminus \cos \left(x\right) = - \setminus \frac{1}{2} : \setminus \quad x = \setminus \frac{2 \setminus \pi}{3} + 2 \setminus \pi n , \quad x = \setminus \frac{4 \setminus \pi}{3} + 2 \setminus \pi n$

Solutions for the range $0 \setminus \le x \setminus \le 2 \setminus \pi$
$x = \setminus \frac{2 \setminus \pi}{3} , x = \setminus \frac{4 \setminus \pi}{3}$

Finally combining all the solutions,

$x = \setminus \frac{2 \setminus \pi}{3} , x = \setminus \frac{\setminus \pi}{4} , x = \setminus \frac{3 \setminus \pi}{4} , x = \setminus \frac{5 \setminus \pi}{4} , x = \setminus \frac{4 \setminus \pi}{3} , x = \setminus \frac{7 \setminus \pi}{4}$