Periodic signals complete repetitive cycles. Each point within a cycle can be described with cyclical units – either degrees or radians. A cycle in units of degrees starts at {0}^{\circ} and finishes at {360}^{\circ}. Equivalently, a cycle in units of radians starts at 0 and finishes at 2\pi. Half a rotation is {180}^{\circ} or {\pi} radians. One quarter of a rotation is {90}^{\circ} or \frac{\pi}{2} radians, etc.


One way to describe and visualize a cycle is to consider a circle, with each point on the circle representing a corresponding point in a cycle. The units of degrees (or radians) are based on an angular rotation of the circle relative to the start of a cycle.


Let’s place this circle on the cartesian axes, centered at the origin. By convention, the horizontal axis will be called the x-axis and the vertical axis will be called the y-axis. If the radius of the circle is 1 with the start of a cycle at the coordinates (1,0), then it is called the Unit Circle.


For an angular rotation, \phi, the trigonometric function \sin(\phi) represents the vertical, y-value for the circle.


When I learned trigonometry in school, trigonometric functions were defined from a right triangle. Therefore, another method to consider is a triangle created by the angular rotation, \phi, relative to the start of a cycle. For the triangle, one point is at the origin, another point is the intersecting point on the circle, and the last point is where a perpendicular line forms a right angle at the x-axis. The following diagram illustrates the Unit Circle and \sin(\phi).




A closely related function to \sin(\phi) is \cos(\phi). This function represents the horizontal, x-value for the circle. Using the previous triangle example, \cos(\phi) = \frac{adjacent}{hypotenuse}Note: for the Unit Circle, the hypotenuse will equal the circle’s radius of 1.


Every point on the Unit Circle can be given by the x and y coordinates, and can be calculated from the angular rotation, (\cos(\phi),\sin(\phi)). The following diagram illustrates several points in the relationship between cartesian coordinates and angular rotation.

Unit Circle


Next, let’s follow the path of rotation through a cycle. Focus on the value of the y coordinate (\sin(\phi)). At the start of a cycle, y = 0. During angular rotation, will increase to 1, return to 0, decrease to -1, and return to 0. The following diagram compares angular rotation on the Unit Circle to a plot of \sin(\phi) versus \phi.


Sine Angle


Now consider the result of completing a cycle while rotating at a constant rate. In the following animation, the green circle represents the cycle. The green dot on the green circle shows the angular rotation changing at a constant rate. The blue dot above the circle traces the x-value (\cos(\phi)) during a cycle. The red dot left of the circle traces the y-value (\sin(\phi)) during a cycle.




The time it takes for a dot to complete a cycle is the period, \tau. The number of cycles per second is the frequency, f = \frac{1}{\tau} Hz.


In conclusion, the trigonometric sine function (and also cosine function) represents position in one dimension due to cyclical rotation at a constant rate and with a constant period. Given a speaker cone travels in one dimension to produce longitudinal waves, the sine function can be used to create a signal with a single frequency.


Now let’s use this sine function as the basis to synthesizing a sine wave.


Photo credits: Sine Function, Unit Circle, Waveform, GIF

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