The sine of 0 degrees (sin 0°) is equal to 0 because of its relationship to the unit circle and the definition of the sine function.
Understanding the Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The sine of an angle is defined as the y-coordinate of the point where the terminal side of that angle intersects the unit circle.
Determining sin 0°
- To find sin 0°, we visualize a 0° angle. This angle lies along the positive x-axis.
- The point where the 0° angle intersects the unit circle is (1, 0).
- The sine of an angle is the y-coordinate of this intersection point.
- Since the y-coordinate is 0, we have sin 0° = 0.
| Angle (Degrees) | x-coordinate | y-coordinate (sin) |
|---|---|---|
| 0° | 1 | 0 |
Why the Y-coordinate Matters
- The definition of the sine function in the context of the unit circle directly ties it to the vertical (y-axis) component of the point.
- When the angle is 0°, there's no vertical displacement, meaning the y-coordinate and thus, the sine value is 0.
- As the angle increases, the y-coordinate and sine value increases until 90 degrees, where the point is at (0,1) and the sine value is 1.
Therefore, the reason sin 0 is 0 is because it corresponds to the y-coordinate of the point on the unit circle at 0 degrees, and that y-coordinate is 0.
