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The cosine function, often written as \(\text{cos}(θ)\), measures the horizontal distance of an angle measured from the unit circle. This trigonometric function plays a crucial role in solving equations that involve periodic behavior. The key properties of the cosine function include:

• Its values range from -1 to 1.
• It is periodic with a period of \(2Ï€\), meaning \(\text{cos}(θ + 2kÏ€) = \text{cos}(θ)\) for any integer \( k \).
• It is an even function, so \( \text{cos}(-θ) = \text{cos}(θ) \).

When solving an equation like \(\text{cos}\frac{x}{2} = \frac{1}{2}\), you'll need to use these properties to identify all possible solutions.

Principal angles are the specific values of an angle between 0 and \(2π\) where the trigonometric functions take common values. For the cosine function, principal angles correspond to well-known outputs such as \( \cos(θ) = \frac{1}{2}\), which happens at \(\frac{π}{3}\) and \(-\frac{π}{3}\). When dealing with an equation like \(\text{cos}\frac{x}{2} = \frac{1}{2}\), it's essential to first identify the principal angles. Once you know these angles, you can generate other solutions by considering the periodic nature of the cosine function.

General solutions are expressions that incorporate all possible solutions to a given equation by accounting for the periodic nature of trigonometric functions. For the equation \( \text{cos}\frac{x}{2} = \frac{1}{2}\), once you identify the principal angles, say \( \frac{π}{3} \), the next step involves generating a formula to find all solutions. Since cosine is periodic with a period of \(2π\), you add \(2kπ\) (where \k\ is any integer) to the initial angle. Therefore, you set up:

\(\frac{x}{2} = \frac{π}{3} + 2kπ\)

\(\frac{x}{2} = -\frac{π}{3} + 2kπ\)

Solving for \( x \) gives:

\ x = \frac{2π}{3} + 4kπ\

\ x = -\frac{2π}{3} + 4kπ\

The general solutions thus integrate all possible values of \( x \) that satisfy the original equation.