Sinusoidal

The term sinusoidal is used to describe a curve, referred to as a sine wave or a sinusoid, that exhibits smooth, periodic oscillation. It is named based on the function y=sin(x). Sinusoids occur often in math, physics, engineering, signal processing and many other areas.

Graph of y=sin(x)

Below are some properties of the sine function:

Graphing sinusoids

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.

y = A·sin(B(x-C)) + D

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

Examples:

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.


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 are 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.

Example:

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 .
    C=

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,

and

would result in the same curve.