The sine of pi (sin π) is zero because of its position on the unit circle.
Understanding the Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. In this context:
- Sine (sin) represents the y-coordinate of a point on the unit circle.
- Cosine (cos) represents the x-coordinate of the point.
Pi and its Position
- The angle π (pi) radians, which is equivalent to 180 degrees, represents a point that lies on the negative x-axis of the unit circle.
Why sin(π) = 0
- Since π lies on the negative x-axis, the y-coordinate at that point is 0.
- As stated in the reference, "Since pi lies on the negative x-axis, the final value of sin pi is 0".
Summary
| Concept | Explanation | Value |
|---|---|---|
| Unit Circle | Circle with radius 1 | |
| sin(θ) | y-coordinate of point on unit circle | |
| π | Angle at 180 degrees (negative x-axis) | |
| sin(π) | y-coordinate when the angle is π radians | 0 |
Therefore, because the sine function corresponds to the y-coordinate, and the angle of π radians results in a point on the unit circle with a y-coordinate of 0, sin(π) is zero.
