The term sinusoidal is used to describe a curve, referred to as a sine wave or a sinusoid, that exhibits smooth, periodic oscillation. Sinusoids occur often in math, physics, engineering, signal processing and many other areas.

Sinusoidal graph

The term sinusoid is based on the sine function y = sin(x), shown below. Graphs that have a form similar to the sine graph are referred to as sinusoidal graphs.

Notice the periodic nature of the sine graph. It repeats every 2π and has smooth curves. Below are some of the properties of the sine function:

The above quantities are only relevant for the function y = sin(x). While sinusoidal graphs will take on the same form as y = sin(x), the quantities describing the graph, such as amplitude, domain, and range, can vary significantly. This in turn affects how the graph will look. It is also worth noting that the cosine function is a sinusoidal graph, as it is simply the sine function with a horizontal shift.

Graphing sinusoidal functions

Most applications cannot be modeled using y = sin(x), and require modification. The equation below is the generalized form of the sine function, and can be used to model sinusoidal functions. This equation may also be colloquially referred to as the sinusoidal function formula or sinusoidal function equation.

Sinusoidal function formula

y = A·sin(B(x-C)) + D

where A, B, C, and D are constants such that:

Consider the following example.


1. Graph y = 3sin(2x)

  • Period:
  • Amplitude: |A| = |3| = 3
  • C = 0, so there is no phase shift
  • D = 0, so there is no vertical shift

Two periods of the graph are shown below. The graph of y = sin(x) is also shown as a reference.

Note how the various quantities affect the shape of the sinusoidal graph as compared to the base graph, y = sin(x).

2. Graph y = 2sin(x - ) + 3.

  • Period:
  • Amplitude: |A| = |2| = 2
  • C = , so the graph shifts right
  • D = 3, so the graph shifts up 3

The graph is shown below.

Equation of a sinusoidal curve

Given the graph of a sinusoidal function, we can write its equation in the form y = A·sin(B(x - C)) + D using the following steps.


Write an equation for the sinusoidal graph below.

  • The maximum value of the graph is 3 and the minimum value is -1, so the equation of the midline is,
  • The sinusoid has maximum at y = 3, and D = 1, so
    A = 3 - 1 = 2
  • There is a maximum at x=. The next maximum after that is at x= so the period is .
  • The first point at which the sinusoid intersects the line y=1 that precedes a local maximum is .

Substituting all of these into the generalized form of the sine function:

Because of the periodic nature of a sinusoid, the equation for a sinusoidal curve is not unique. We could have found different points for C, such as (, 1) or (, 1), and their equations,


would result in the same curve.