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The cosine function, denoted as \(\cos\), is one of the fundamental trigonometric functions. It is used to determine the horizontal coordinate of a point lying on the unit circle for a given angle. In a right-angled triangle, the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse. For an angle \(x\), mathematically it's represented as \(\cos(x)\).

In solving trigonometric equations like \(2 \cos x+1=0\), understanding the cosine function properties is crucial. Our equation is manipulated to isolate \(\cos\) and reduce it to a familiar form \(\cos x = -\frac{1}{2}\). Knowing the values which \(\cos\) assumes at standard angles allows us to determine the specific angles \(x\) that satisfy the given equation.

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It's an invaluable tool in trigonometry for understanding the trigonometric functions and their periodicity. Points on the unit circle correspond to angles and their trigonometric functions values.

When we refer to angles like \(2\pi/3\) and \(4\pi/3\), which solve the equation \(\cos x = -\frac{1}{2}\), we’re essentially seeking points on the unit circle where the x-coordinate equals \( -\frac{1}{2} \). These points are found by drawing a horizontal line through the y-axis at \( -\frac{1}{2} \) and noting the intersections with the unit circle. These intersections indicate the angles from the positive x-axis where the cosine value meets the required condition.

A periodic function is one that repeats its values at regular intervals or periods. The cosine function is an example of a periodic function with a period of \(2\pi\). This means that the function completes a cycle and repeats its values every \(2\pi\) radians. In essence, if you have found an angle \(x\) that satisfies our trigonometric equation, you can add \(2\pi\) to \(x\) any integer number of times (denoted as \(2n\pi\), where \(n\) is an integer) and you will find another solution to the equation.

The general solution of \(x = 2n\pi \pm 2\pi/3\) for our exercise is derived from understanding this periodic nature. Because cosine is periodic, we know that the angle solutions we found on the unit circle are not the only ones — they repeat every \(2\pi\) radians. Hence, the equation’s solution includes all angle measures that differ from the principal angles by full cycles of \(2\pi\).