91Ó°ÊÓ

The \textbf{sine function}, denoted as \(\text{sin}(\theta)\), comes from trigonometry. It links an angle \(\theta\) to a ratio in a right triangle, specifically: the length of the side opposite the angle divided by the hypotenuse. In the context of the unit circle (a circle with radius 1 centered at the origin), the sine function gives the y-coordinate of a point corresponding to the angle \(\theta\).
The sine function repeats its values in a cyclical manner every \(2\pi\) radians which is one full revolution of the circle. This repeating behavior is called 'periodicity'. The core characteristic to remember is:

• \(\text{sin}(\theta) = \text{sin}(\theta + 2\pi)\)

where \(\theta\) can be any angle. Prime angles to remember are:

• \(\text{sin}(0) = 0\)
• \(\text{sin}(\pi/2) = 1\)
• \(\text{sin}(\pi) = 0\)
• \(\text{sin}(3\pi/2) = -1\)
• \(\text{sin}(2\pi) = 0\)

In the given problem, \(\sin(\pi x) = 1/2\), we utilize these core characteristics to determine the possible angles where the sine function equals \(1/2\).

The \textbf{unit circle} is a fundamental tool in trigonometry. Its center is at the origin \(0,0\) and its radius is 1 unit. Every point \((x,y)\) on the unit circle corresponds to the coordinates \((\cos(\theta), \sin(\theta))\) for a given angle \(\theta\) formed with the positive x-axis.
Let's focus on the sine values of certain known angles:

• At \(\theta = \pi/6\) or 30 degrees, the y-coordinate is \(1/2\), meaning \(\sin(\pi/6) = 1/2\).
• At \(\theta = 5\pi/6\) or 150 degrees, the y-coordinate is also \(1/2\), meaning \(\sin(5\pi/6) = 1/2\).

The unit circle helps visualize these values. We can see that the positions \(\pi/6\) and \(5\pi/6\) on the circle give the correct sine value. The \