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The cosine function is one of the basic trigonometric functions representing the ratio of the adjacent side to the hypotenuse of a right-angled triangle. Mathematically, it is defined for an angle \( \theta \) as \( \cos(\theta) \). The cosine function is even, which means that \( \cos(\theta) = \cos(-\theta) \), and it oscillates between -1 and 1, inclusive.

When solving trigonometric equations, understanding the cosine function is crucial. For instance, in the given equation \(2 \cos x - 1 = 0\), we're looking for values of \(x\) that make the cosine function equals to 0.5. Within the range of \(0 \leq x \leq 2\pi\), the cosine function reaches the value of 0.5 at specific standard angles which are commonly found using the unit circle or trigonometric tables.

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is a fundamental tool in trigonometry because every point on the unit circle corresponds to an angle, measured from the positive x-axis, with coordinates \((\cos(\theta), \sin(\theta))\). This makes it an excellent resource for visualizing and solving problems involving trigonometric functions.

In the context of our problem, the unit circle helps to identify angles for which the cosine value is 0.5. These points are located where the horizontal coordinate (cosine value) is 0.5; using the unit circle, one can see that this occurs at \(\theta = \pi/3\) and \(\theta = 5\pi/3\), which are the only two solutions in the interval \([0, 2\pi]\).

A solution set in mathematics is a collection of all possible values that satisfy a given equation or inequality. In the domain of trigonometry, the solution set for an equation such as \((2 \cos x - 1)(3 + 2 \cos x) = 0\) includes all the angles \(x\) which make the entire equation true. Given that the cosine function has a limited range, not all real numbers can be solutions.

In our problem, after determining the cosine values that solve each factor in the equation, we determined that \(\cos x = 0.5\) has solutions and \(\cos x = -1.5\) does not, since it falls outside the range of the cosine function. Consequently, the solution set for the given interval is the set of all angle measurements that result in the appropriate cosine values, which for this equation is \(\{ \pi/3, 5\pi/3 \}\). This illustrates the importance of understanding the range and behavior of trigonometric functions when seeking the solution set for trigonometric equations.