Question

How do you solve $\cos 2x = \cos x$ from $0$ to $2\pi $?

Answer 1

Hint:
For solving such a type of question first we will use a formula which includes both $\cos x$ and $\cos 2x$. After that we will put the value of $\cos 2x$ to $\cos x$, because they both are equal from $0$ to $2\pi $.

Complete step by step solution:
So for solving such a type of question we have to use an equation in which we have both $\cos 2x$ and$\cos x$.
Here, best formula which include $\cos 2x$and $\cos x$ is given below,
$ \Rightarrow \cos 2x = 2{\cos ^2}x - 1$
Now, in this equation we substitute values of $\cos 2x$which is $\cos x$ for the interval $0$ to $2\pi $,
So we get,
$ \Rightarrow \cos x = 2{\cos ^2}x - 1$
After rearranging above equation we get,
$ \Rightarrow 2{\cos ^2}x - \cos x - 1 = 0$
Now, we have to solve above equation and also have to find cos values,
$ \Rightarrow 2{\cos ^2}x - 2\cos x + \cos x - 1 = 0$
Now in first two terms we can get common $2\cos x$, so after that we get,
$ \Rightarrow 2\cos x(\cos x - 1) + 1(\cos x - 1) = 0$
$ \Rightarrow (2\cos x + 1)(cosx - 1) = 0$
Now, $2\cos x + 1 = 0$ or $\cos x - 1 = 0$
$\cos x = - \dfrac{1}{2}$ 0r $\cos x = 1$
So, if $\cos x = - \dfrac{1}{2}$ then x for interval $0$ to $2\pi $ is $\dfrac{{2\pi }}{3},\dfrac{{4\pi }}{3}$

And if $\cos x = 1$ then x for interval $0$ to $2\pi $ is $0,2\pi $.

Note:
For solving such types of questions first you must have to understand the question and find (try to remember) the best way to solve it. you should first write-down what has been given to you and what you have to find. Then use formulas to solve it.