Question

Solve the equation. (Enter your answers as a comma-separated list. Use n as an arbitrary integer. Enter your response in radians.)

Answer 1

ANSWER

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

EXPLANATION

To solve this equation, first, we have to subtract 4 from both sides of the equation,

$\begin{gathered} 7\cos x+4-4=-\cos x-4 \\ \\ 7\cos x=-\cos x-4 \end{gathered}$

Then, add cos x to both sides,

$\begin{gathered} 7\cos x+\cos x=\cos x-\cos x-4 \\ \\ 8\cos x=-4 \end{gathered}$

Divide both sides by 8,

$\begin{gathered} \frac{8\cos x}{8}=\frac{-4}{8} \\ \\ \cos x=-\frac{1}{2} \end{gathered}$

If we look at the unit circle, we will find that there are two angles whose cosine is -1/2,

And these two angles only repeat every 2π radians.

Hence, the solutions to this equation are:

$x=\frac{2\pi}{2}+2πn,x=\frac{4π}{3}+2πn$