Sinusoidal Waveform Equation

Sinusoidal waveform is a waveform that oscillates periodically or has a frequency and fulfills the sine calculation. This waveform has a shape of S, going up and down periodically with positive and negative amplitude.

Of course, not only sine function, we can make a sinusoidal waveform with cosine function. Its shape will be the same but its starting point will be different.

If you have learnt about sine and cosine function, you will realize the cosine function is shifted 90o to sine function.

Take an example when the time is x = 0 (X-axis).

At the exact time frame, the sine will start at 0 while the cosine will start at 1. Hope this helps before advancing further.

What is an Electric Signal?

Sinusoidal alternating current and Sinusoidal voltage are types of electrical signal. This signal has an electrical quantity, may it be voltage or current. We can use this to rally information from a point to other points through a medium such as electromagnetic, wire, etc.

But utilizing this signal is not simple as it sounds, we need to analyze the waveform, amplitude, distance, medium, and behavior of the transmitted signal. This graph used to analyze the waveform is also known as a representation of sinusoidal waveforms.

A point to remember, a signal used to deliver an information consists of message signal and carrier signal. Just as it sounds, the message signal is the signal containing the information, while the carrier signal is the necessary energy for the flow of the signal.

Representation of Sinusoidal Waveforms in AC Circuit

Now we will talk about the sinusoidal waveform. A periodic signal in an AC circuit with a function of sine or cosine function in trigonometry is called representation of sinusoidal waveform in AC circuit. This sinusoidal signal is also known as sinusoid, for short.

Just as mentioned above, this waveform has the shape of letter S in a period of time. As we observe deeper, this S-shape consists of a pair of arcs or half circles with positive and negative amplitude.

For better understanding, while drawing the sinusoidal waveform we will use a two dimensional graph consisting of X-axis and Y-axis.

Sinusoidal waveform has an amplitude which indicates the maximum value or its peak value.

The X-axis will be the time axis and the Y-axis will be the amplitude axis.

As we have read above, we can use cosine or sine function to form a sinusoid only with different starting points.

⍵ is the angular frequency in rad/s (radians per second)
⍵t is the argument of the sinusoid

The sine function will produce a sine wave signal while the cosine function will produce a cosine wave signal.

When t = 0 at X-axis, then

The sine wave signal will have a starting point at 0 at Y-axis. Its amplitude value will go up to 1 when ⍵t = 90 then go down to zero again when ⍵t = 180 then go down further to -1 when ⍵t = 270 and go up to zero when ⍵t = 360.

The cosine wave signal will have a starting point at 1 at Y-axis. Its amplitude value will go down to 0 when ⍵t = 90 then go down to -1 when ⍵t = 180 then go up to 0 when ⍵t = 270 and go up again to 1 at ⍵t = 360.

The sinusoidal waveform has continuous value depending on the ⍵t. It has an instantaneous value at a particular timestamp. Since it depends on the time, we can conclude that a sinusoidal waveform is a function of time and written as a time function f(t).

What is a Radians in Sinusoidal AC

Radians or rad for short, mathematically, is the unit of angle in the International System of Units and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius.

1 rad if converted into degree, it will be 

180 / π = 57.296o

Or since a full circle 360o is equal to 2π then

Or we can use the equations below

With a few example:

AC Sinusoidal Waveform

Observe the example of Alternating Current Sinusoidal Waveform below.

sinusoidal waveform 1

From the sinusoidal waveforms above, we write the sinusoidal function of ac sine wave


v(t) = time function

Vm = amplitude of the sinusoid

⍵ = angular frequency in rad/s

⍵t = argument of the sinusoid 

T = time period (s)

π = ratio of the circumference of a circle to its diameter

Why do we use π?

Because this variable represents the full circle cut in half to form a sinusoidal waveform.

From the waveform above, we can see the repeat cycle of a sinusoid every T seconds, thus the T is what we called the period of the sinusoid.

Comparing those two waveform we can conclude that ⍵T = 2π. Thus

We can notice that v(t) has repetitive value every T seconds, with t replaced by t + T


A periodic function is one that fulfills f(t) = f(t + nT), for all t and for all integers n.

Please do not think t and T are the same. The t indicates the time while the T indicates the period or the time required for one cycle to complete or the number of seconds per cycle.

The opposite or reciprocal value of period (T) is frequency (f) measured in Hertz or Hz. The frequency indicates the number of cycles per second.

The higher the frequency means more cycles occur in a second or the less time it needs to complete one cycle.


The sinusoidal wave equation is the multiplication of the voltage amplitude with the sine function of the sum of arguments of the sinusoid with the phase.

Just as stated above, the ⍵t is the argument of the sinusoid and the ∅ is the phase. Both can be in radians or degrees.

For a more complete sinusoidal wave equation, we can use either one below.

Sinusoidal waveform equation with a phase will show whether the waveform lags or leads based on the radians or degree. Observe the sinusoidal waveform below.

sinusoidal waveform 2

With the equation above, we can see that:

  • v2 leads v1 by ∅ (it starts before ⍵t = 0)
  • v1 lags v2 by ∅ (it starts at ⍵t = 0)

AC Sine Wave Equation Examples

We will learn how to use the AC sine wave equation with a known amplitude, phase, periode, and frequency in a function below.

From the sine function above, we get some variables:

v(t) = sinusoidal ac

Amplitude = 12

Phase (∅) = 10o

Angular frequency (⍵) = 50 rad/s

With the known angular frequency, we can calculate the period time and frequency

Why Sinusoidal Waveform is Used for AC

It is not difficult now to explain why sinusoidal waveforms are important in electrical and electronic aspects and used for AC circuits. This waveform is the natural representation of alternating current. This waveform is oscillating endlessly when supplied to the circuit with a specific frequency, amplitude, and phase.

This oscillating waveform has its own mathematical function that can be used in addition, subtraction, multiplication, and division.

So why sinusoidal waveform?

We can see this in an AC generator while operating. The circular movement to the generator represents the sinusoidal waveform if both positive polarity and negative polarity is fused together to form a perfect circular shape.

Let’s say we rotate the shaft of an AC generator 180o, it will produce a positive period of sine wave and if we continue to rotate the shaft from 180o to 360o, it will produce the negative period of sine wave.

This process is repeated and produces the oscillating sinusoidal waveform.

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