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The sine function is a fundamental aspect of trigonometry. It is denoted by \( \sin \theta \) and describes the y-coordinate of a point on the unit circle corresponding to an angle \( \theta \). This function varies between -1 and 1 as it moves around the unit circle.
It's essential to grasp that the sine function is periodic, which means it repeats its values in regular intervals. Specifically, the period of the sine function is \( 2\pi \). This implies that sine returns to its starting point after every \( 2\pi \) radians.
For the equation \( \sin \beta = -\frac{\sqrt{3}}{2} \), knowing the specific interval where the sine of an angle equals this value is key. These specific angles play a crucial role in solving trigonometric equations.

The unit circle is an indispensable tool in trigonometry that helps us understand the behavior of trigonometric functions like sine. It is a circle with a radius of 1, centered at the origin on the coordinate plane.
Each point on the unit circle corresponds to a unique angle measured in radians from the positive x-axis. By definition, the sine of an angle is the y-coordinate of its corresponding point on the unit circle. Hence, understanding the unit circle helps us quickly find sine values for common angles.
For example, for \( \sin \beta = - \frac{\sqrt{3}}{2} \), the corresponding angles on the unit circle are \( 4\pi/3 \) and \( 5\pi/3 \). These angles occur in the third and fourth quadrants of the circle where sine values are negative.

General solutions in trigonometry refer to all possible angles that satisfy a given trigonometric equation. Since trigonometric functions are periodic, they have infinitely many solutions. To express all these solutions, we use a general formula that incorporates the periodic nature of the circle.
In the case of \( \sin \beta = -\frac{\sqrt{3}}{2} \), the general solution is derived from the specific solutions \( \beta = 4\pi/3 \) and \( 5\pi/3 \). To account for the full set of solutions given the periodicity of sine, we add \( 2k\pi \), where \( k \) is any integer, to these specific angles:

• \( \beta = 4\pi/3 + 2k\pi \)
• \( \beta = 5\pi/3 + 2k\pi \)

This expression covers all rotations around the unit circle, ensuring that every possible solution is included.