The sine of pi is zero because pi radians corresponds to 180 degrees, which is the point on the unit circle where the y-coordinate is zero.
Understanding Sine and the Unit Circle
- Sine Function: The sine function (sin) in trigonometry is a ratio that relates the length of the opposite side of a right triangle to the length of its hypotenuse.
- 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.
- Sine as Y-coordinate: On the unit circle, the sine of an angle is represented by the y-coordinate of the point where the angle intersects the circle.
Why sin(pi) = 0
- Angle of pi: Pi radians is equivalent to 180 degrees.
- Point on the unit circle: When you rotate 180 degrees counterclockwise from the positive x-axis on the unit circle, you end up at the point (-1, 0).
- Sine as y-coordinate: The y-coordinate of this point is 0, which means the sine of pi (or 180 degrees) is 0.
Visual Representation
Imagine a right triangle formed by the radius of the unit circle, the x-axis, and the line segment connecting the origin to the point where the angle intersects the circle. When the angle is 180 degrees, the opposite side of the triangle is zero, making the sine of the angle (opposite side / hypotenuse) equal to zero.
