We have the expression:
[tex]2\cos x+1=0[/tex]
To find x, first, we solve for cosx by subtracting 1 to both sides of the equation:
[tex]\begin{gathered} 2\cos x+1-1=-1 \\ 2\cos x=-1 \end{gathered}[/tex]
Then, divide both sides by 2:
[tex]\begin{gathered} \frac{2\cos x}{2}=-\frac{1}{2} \\ \cos x=-\frac{1}{2} \end{gathered}[/tex]
Here we have that the value of x is such that the cosine of that value is equal to -1/2.
We can use the inverse cosine to solve for x:
[tex]x=\cos ^{-1}(-\frac{1}{2})[/tex]
And the result is:
[tex]x=2.094[/tex]
And we look in our options which one has the corresponding decimal value:
[tex]\begin{gathered} \frac{\pi}{3}=1.047 \\ \frac{2\pi}{3}=2.094 \\ \frac{4\pi}{3}=4.189 \\ \frac{5\pi}{3}=5.236 \end{gathered}[/tex]
And we can see that 2pi/3 is equal to the value that we got for x, so:
[tex]x=\frac{2\pi}{3}[/tex]
Answer:
[tex]x=\frac{2\pi}{3}[/tex]
