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.

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 $\phi$.

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