91Ó°ÊÓ

The sine function, represented as \(\text{sin} \theta\), is a fundamental concept in trigonometry. It relates to the y-coordinate of a point on the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. When we talk about \(\text{sin} \theta\), we are referring to the vertical distance from the x-axis to a point on the unit circle.
The sine function produces a wave-like graph that oscillates between -1 and 1. In simpler terms, it's like the height of a wave above or below a neutral line. For any angle, \(\theta\), \(\text{sin} \theta\) gives us the height of the point on the circle relative to the horizontal axis.
In this exercise, we are given \(\text{sin} \theta = 0\), meaning we need to find all angles where the y-coordinate is zero. These are the points where the wave crosses the horizontal axis.

The unit circle is a critical tool in trigonometry. It is a circle with a radius of exactly one unit, centered at the origin of a coordinate plane (0, 0). This simple geometric shape helps to define the sine, cosine, and tangent functions.
In the context of the given problem, finding \(\text{sin} \theta = 0\) means locating points on the unit circle where the y-coordinate is zero. These points lie along the x-axis.
Two important angles to consider on the unit circle are 0 degrees and 180 degrees. At these angles, the point on the unit circle touches the x-axis, meaning the y-coordinate or \(\text{sin} \theta\) is zero. Hence, \(\theta\) values of 0 degrees and 180 degrees are solutions to \(\text{sin} \theta = 0\).

To solve the equation \(\text{sin} \theta = 0\), we need to find all possible angles \(\theta\) within a full rotation of the circle that meet the given condition. An angle that satisfies the constraint \(\text{sin} \theta = 0\) must lie on the x-axis of the unit circle. Let's break down the solution:

• At 0 degrees (\theta = 0^{\text{\textdegree}})
• At 180 degrees (\theta = 180^{\text{\textdegree}})

These are the only points where \(\text{sin} \theta\) equals zero within the interval \(0^{\text{\textdegree}} \leq \theta \< 360^{\text{\textdegree}}\). Therefore, the complete set of solutions is \(\theta = 0^{\text{\textdegree}} \text{and} \theta = 180^{\text{\textdegree}}\). These solutions fall well within the specified range, ensuring no angles are missed.
This approach can be extended to solve other trigonometric equations involving different trigonometric functions and their respective conditions on the unit circle.