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

`x=(2\pi)/(2)+2\pi n, x=(4\pi)/(3)+2\pi n`

Step-by-step 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} (8\cos x)/(8)=(-4)/(8) \\ \\ \cos x=-(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=(2\pi)/(2)+2πn,x=(4π)/(3)+2πn`