There is a deep mathematical relationship between sine waves and circles. This connection can be understood through trigonometry and the unit circle.
In trigonometry, the sine function (sin) relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. However, the sine function can also be extended beyond right triangles to any angle in the unit circle.
The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) in a Cartesian coordinate system. By associating angles with points on the unit circle, we can define the sine of an angle as the y-coordinate of the corresponding point on the unit circle.
As an angle rotates counterclockwise around the unit circle, the y-coordinate of the corresponding point on the circle traces out a wave-like pattern. This pattern is a sine wave. The sine wave represents the projection of the rotating point onto the vertical axis.
Therefore, the connection between sine waves and circles arises from the relationship between the angle of rotation and the coordinates on the unit circle. Sine waves are a graphical representation of the circular motion and can be described mathematically using trigonometric functions such as sine and cosine.