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:

\[\cos x = - \frac{1}{2} \]

\[ \Rightarrow \cos x = \cos\frac{2\pi}{3}\]

\[ \Rightarrow x = 2n\pi \pm \frac{2\pi}{3}\]
By putting n = 0 and n = 1 in the above equation, we get:

\[x = \frac{2\pi}{3}\] or \[x = \frac{2\pi}{3}\]  in the interval 

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