Sinusoidal function
From Calculus
Contents
Definition
As a linear transform of the sine function
The term sinusoidal function refers to a function of the form where
and
are linear functions and
is the sine function. Specifically, it is a function of the form:
Here:
-
is the mean value about which the function is oscillating, i.e., the graph of a function looks like a scaled sine function about the horizontal line
.
-
is the amplitude of oscillations, i.e., the function oscillates between a minimum value of
and a maximum value of
.
-
is the angular frequency parameter and controls the period of oscillations, which is given by
.
-
is a phase parameter that roughly describes the head start of the function relative to a function that starts at its mean value at
.
As a linear combination of sine and cosine functions
The term sinusoidal function can be used for a function of the form:
Conversion between the two versions
Here's how we convert the linear combination version to the linear transform version:
-
remain the same.
- Set
.
- Set
as an angle so that
and
.
is uniquely determined up to additive multiples of
.
Here's how we convert the linear transform version to the linear combination version:
-
remain the same.
-
.
-
.
Examples
Function | How it's a sinusoidal function in the linear transform sense | How it's a sinusoidal function in the linear combination sense |
---|---|---|
sine function | ![]() ![]() |
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cosine function | ![]() ![]() |
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sine-squared function ![]() |
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cosine-squared function ![]() |
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