The sine of 180 degrees (sin 180°) is equal to 0 because of its position on the unit circle and trigonometric properties.
Understanding the Unit Circle
To understand why, let's look at 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 counter-clockwise from the positive x-axis.
- The coordinates of any point on the unit circle are given by (cos θ, sin θ), where θ is the angle from the positive x-axis.
Analyzing 180°
- An angle of 180° places the corresponding point on the unit circle exactly on the negative x-axis.
- The coordinates of this point are (-1, 0).
- Since sine corresponds to the y-coordinate, sin 180° = 0.
Explanation from Reference
The provided reference directly states, "Since 180° lies on the negative x-axis, the final value of sin 180° will be 0." This explanation aligns perfectly with the unit circle interpretation explained above. It also mentions a trigonometric identity: sin(180° - 180°) = sin 0°. While this identity is true, it doesn't directly explain why sin 180° is zero; it only shows it’s equivalent to sin 0, which is also zero, as the point lies on the x axis.
Summary
| Angle | Location on Unit Circle | x-coordinate (cos) | y-coordinate (sin) |
|---|---|---|---|
| 0° | Positive x-axis | 1 | 0 |
| 90° | Positive y-axis | 0 | 1 |
| 180° | Negative x-axis | -1 | 0 |
| 270° | Negative y-axis | 0 | -1 |
Therefore, sin 180° is 0 because the y-coordinate of the point corresponding to 180° on the unit circle is 0, and from a trigonometric point of view, it lies on the x-axis.
