Question

A. pi/3b. 2pi/3c. 4pi/3d. 5pi/3.Find the solution of each equation on the interval ( 0, 2pi).

Answer 1

We have the expression:

$2\cos x+1=0$

To find x, first, we solve for cosx by subtracting 1 to both sides of the equation:

$\begin{gathered} 2\cos x+1-1=-1 \\ 2\cos x=-1 \end{gathered}$

Then, divide both sides by 2:

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

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:

$x=\cos ^{-1}(-\frac{1}{2})$

And the result is:

$x=2.094$

And we look in our options which one has the corresponding decimal value:

$\begin{gathered} \frac{\pi}{3}=1.047 \\ \frac{2\pi}{3}=2.094 \\ \frac{4\pi}{3}=4.189 \\ \frac{5\pi}{3}=5.236 \end{gathered}$

And we can see that 2pi/3 is equal to the value that we got for x, so:

$x=\frac{2\pi}{3}$

Answer:

$x=\frac{2\pi}{3}$