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The cosine function, denoted as \(\text{cos} \theta\), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. The cosine function is defined as the x-coordinate of the point on the unit circle at a given angle \(\theta\). This function has a range of \([-1, 1]\) and a period of \(2\text{Ï€}\), meaning that its values repeat every \(2\text{Ï€}\) radians. Hence, if \(\text{cos} \theta = a\), then the solutions occur at various intervals determined by the periodic nature of the cosine function.
The general solution for a cosine equation \( \text{cos} \theta = a \) can be formulated as follows:

• \( \theta = 2n \text{Ï€} + \text{arccos}(a) \)
• \( \theta = 2n \text{Ï€} - \text{arccos}(a) \)

Here, \(n\) is any integer, indicating that the function repeats every \(2\text{Ï€}\) radians. Understanding this property of the cosine function can help solve trigonometric equations effectively.

The arccosine function, denoted as \(\text{arccos}(x)\), is the inverse of the cosine function. It returns the angle \(\theta\) for which the cosine value is \(x\). For example, \(\text{arccos}\bigg(\frac{1}{2}\bigg)\) gives the angle whose cosine is \(\frac{1}{2}\). The principal value of arccosine lies in the interval \([0, \text{Ï€}]\).

In our problem, \(\text{cos} \theta = \frac{1}{2}\), we need to find \(\text{arccos}\bigg(\frac{1}{2}\bigg)\). We know the angle whose cosine is \(\frac{1}{2}\) in the primary range is \(\frac{\text{Ï€}}{3}\). This arccosine value is crucial in determining the general solutions of the equation. Using this arccosine value, we can set up our sinusoidal general solutions as follows:

• \( \theta = 2n \text{Ï€} + \frac{\text{Ï€}}{3} \)
• \( \theta = 2n \text{Ï€} - \frac{\text{Ï€}}{3} \)

These equations incorporate all possible angles that have the given cosine value, accounting for the periodic nature of the cosine function.

A function is called periodic if it repeats its values at regular intervals or periods. Trigonometric functions like sine, cosine, and tangent fall into this category. The cosine function, specifically, has a period of\(2\text{Ï€}\), which means it repeats every \(2 \text{Ï€}\) radians.

For example, if \( \text{cos} \theta = 1 \), it can be observed again at intervals of \(2 \text{Ï€}\) radians, like \(\theta = 0, 2\text{Ï€}, 4\text{Ï€}, -2\text{Ï€}\), etc. This repetitiveness is essential for determining the general solution of trigonometric equations.

When dealing with equations like \( \text{cos} \theta = \frac{1}{2} \), acknowledging the periodic nature allows us to write the solution in the general form. By correctly incorporating the period, we get:

• \( \theta = 2n \text{Ï€} + \frac{\text{Ï€}}{3} \)
• \( \theta = 2n \text{Ï€} - \frac{\text{Ï€}}{3} \)

These expressions represent all the angles that have the same cosine value, extending to every integer multiple of \(2\text{Ï€}\). This general solution format ensures that all possible angles are captured, reiterating the periodicity trait of the cosine function.