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The sine function is one of the fundamental functions in trigonometry, denoted as \(\text{sin}(\theta)\). It represents the y-coordinate of a point on the unit circle. The function takes an angle \(\theta\) as input and returns a value between \(-1\) and \(+1\). This is because the unit circle has a radius of 1.\(

\)Some key properties of the sine function:

• **Range:** The sine function outputs values between -1 and 1.
• **Domain:** The input \(\theta\) can be any real number.
• **Periodicity:** The sine function repeats its values every \(2\pi\) radians.
• **Symmetry:** The sine function is an odd function, meaning that \(\text{sin}(-\theta) = -\text{sin}(\theta)\).

Understanding these properties can help you solve trigonometric equations more efficiently.

Trigonometric functions like sine and cosine exhibit periodic behavior. This means that their values repeat at regular intervals. For the sine function, this interval is \(2\pi\) radians.\(

\)Because of this periodicity, if \(\theta\) is a solution to \(\text{sin}(\theta) = \text{A}\), then \(\theta + 2k\pi\) is also a solution for any integer \(k\).\(

\)In our example, \(\text{sin}(\theta) = \frac{1}{2}\), we initially find two solutions within one period: \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\). By adding multiples of \(2\pi\), we can get additional solutions like:

• \(\frac{\pi}{6} + 2k\pi\)
• \(\frac{5\pi}{6} + 2k\pi\)

Where \(k\) is any integer. This periodicity is essential for finding the general solution to trigonometric equations.

When solving trigonometric equations, our goal is often to find all possible solutions. For equations involving the sine or cosine functions, this means considering the periodic nature of these functions.\(

\)In the given problem, we start with \(\text{sin}(\theta) = \frac{1}{2}\). Within one period \(0 \text{ to } 2\pi\), the specific solutions are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\).\(

\)To find all possible solutions, we use the periodicity of the sine function, adding \(2k\pi\) to each of the basic solutions:

• \(\theta = \frac{\pi}{6} + 2k\pi\)
• \(\theta = \frac{5\pi}{6} + 2k\pi\)

Here, \(k\) can be any integer. This approach generalizes the solution to cover every possible value of \(\theta\) that satisfies the original equation. This method is fundamental in solving many trigonometric equations systematically.\(

\)For example, turning integers \(k = 0, 1, -1\) yields some specific solutions:

• \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{13\pi}{6}, \frac{17\pi}{6}, -\frac{11\pi}{6}, \text{ and } -\frac{7\pi}{6}\)

These specific cases help illustrate how the general formula covers an infinite number of solutions.