Sine Wave Equation

The equation for a sine wave is widely published, relating to various applications. Therefore, rather than have a detailed explanation of the sine wave equation for each instance it appears on our website, we show it once here.

2 π f t + Θ

Let’s first discuss the venerable pi (π) component. Pi is the ratio of a perfect circle’s circumference divided by its diameter and thus has no units.

Component f is a measure of frequency having units of cycles per second but is more commonly referred to as Hertz. We will retain the cycles/second designation for algebraic illustration purposes. However, this page will probe a little deeper into the subject than is usually covered in academic texts. The goal is that the reader would understand where these components come from and how they arrived rather than follow some craze cookbook.

The component t is a measure of time and has units of seconds.

What units do we end up with when we multiply these three components of this first term of the sine equation’s argument?

2 π f cy/sec t sec = 2 π f cycles

Notice that the time and seconds cancel, leaving us with cycles. To understand this argument, it is necessary to understand that while time is relevant, time drops out, leaving us with a rational number of cycles without time. If we had represented frequency with units of Hertz, the equation could not be simplified to these components.

Because of this difficulty, the component omega (ω) was likely realized within the industry. The term 2πf occurs so frequently in processes over all disciplines of nature that the omega term (ω) was invented to simplify things.

ω = 2πf (cy/sec or radians/sec or Hz)

Radians relate to degrees as

2π radians = 360o or

π radians = 180o

Because of this, the subject equation of this page is often given as:

e = eo sin(2πft + θ) = eo sin(ωt + θ)

This leads to a discussion of what theta (θ) represents in the equation. The above equation shows that theta (θ) will mean the same units as ωt.

Let’s look at an example illustrated in Figure 1.

In the illustration, we have two or two sine wave phases. Note that the two signals cross zero together in periodical precison. From this, we know that they share a standard frequency and may therefore be separated in time (not frequency) by so many degrees of a period or radians of a period. A period is a time that it takes for them to complete one cycle.

The Figure 1 illustration does not give enough information to solve for a frequency, but we know that the two signals are 180o apart. The sine wave equations representing the Figure 1 signals may therefore be given as

eo sin(ωt + π) or

eo sin(ωt + 180o)

Thus, the two signals shown in Figure 1 can be represented by the above equation Phase A has a theta of zero, and Phase B has a theta of 180o. What is important to note is that both share the identical ωt. Theta then becomes a horizontal offset enabling shifting phases within an equation with ease.