Question

Write the solution set of the equation $\left(2\cos x+1\right)\left(4\cos x+5\right)=0$ in the interval [0, 2π].

Answer 1

Given: $(2\cos x+1)(4\cos x+5)=0$
Now, $2\cos x+1=0$ or $4\cos x+5=0$
$\Rightarrow \cos x=-\frac{1}{2}$ or $\cos x=-\frac{5}{4}$
$\cos x=-\frac{5}{4}$ is not possible.
Thus, we have:
$$\begin{gathered} \cos x=-\frac{1}{2} \\ \Rightarrow \cos x=\cos \frac{2\mathrm{\pi }}{3} \\ \Rightarrow x=2n\mathrm{\pi }\pm \frac{2\mathrm{\pi }}{3} \end{gathered}$$
By putting n = 0 and n = 1 in the above equation, we get:

$x=\frac{2\mathrm{\pi }}{3}$ or $x=\frac{4\mathrm{\pi }}{3}$ in the interval $\left[0,2\mathrm{\pi }\right]$
For the other value of n, x will not satisfy the given condition.
∴ $x=\frac{2\mathrm{\pi }}{3}$ and $\frac{4\mathrm{\pi }}{3}$