Question

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

Answer 1

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

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