Sinusoidal Functions (College Board AP® Precalculus): Revision Note

Sinusoidal functions

What is a sinusoidal function?

• A sinusoidal function is any function that can be obtained by applying additive and multiplicative transformations to the sine function $f\left(\theta \right)=\sin \theta$

  • Both the sine function and the cosine function are sinusoidal functions

• The cosine function is a phase shift (horizontal translation) of the sine function

  • $\cos \theta =\sin \left(\theta +\frac{\pi }{2}\right)$

  • i.e. $\cos \theta$ is the same as $\sin \theta$ shifted to the left by $\frac{\pi }{2}$ radians

• This means any sinusoidal function can be written in terms of either sine or cosine

• Sinusoidal functions are used to model a wide range of periodic phenomena

  • They are the most common type of periodic function encountered in this course

Key characteristics of sinusoidal functions

What are the period and frequency of a sinusoidal function?

• The period of a sinusoidal function is the length of one complete cycle

  • For the base functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$, the period is $\mathbf{2}\mathit{\pi }$

• The frequency of a sinusoidal function is the number of complete cycles per unit of input

  • For the base sine and cosine functions, the frequency is $\frac{\mathbf{1}}{\mathbf{2}\mathit{\pi }}$

• The period and frequency are always reciprocals of each other

  • $\text{frequency}=\frac{1}{\text{period}},\text{period}=\frac{1}{\text{frequency}}$

What is the amplitude of a sinusoidal function?

• The amplitude of a sinusoidal function is half the difference between the maximum and minimum output values

  • $\text{amplitude}=\frac{\text{maximum}-\text{minimum}}{2}$

• For the base functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$

  • The maximum value is $1$ and the minimum value is $-1$

  • So the amplitude is $\frac{1-(-1)}{2}=1$

• The amplitude measures the height of the wave above (or below) its midline

What is the midline of a sinusoidal function?

• The midline of the graph of a sinusoidal function is the horizontal line located at the average (arithmetic mean) of the maximum and minimum output values:

  • $\text{midline}:y=\frac{\text{maximum}+\text{minimum}}{2}$

• For the base functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$

  • The midline is $y=\frac{1+(-1)}{2}=0$

• The graph oscillates symmetrically above and below the midline

• The amplitude can also be understood as the distance from the midline to the maximum (or to the minimum)

How does the concavity of a sinusoidal function change?

• As the input values increase, the graph of a sinusoidal function oscillates between concave down and concave up

  • Sections of the graph near a maximum are concave down

    • the graph curves downward, shaped like an upside-down cup

  • Sections of the graph near a minimum are concave up

    • the graph curves upward, shaped like a cup

• The changeover between concave up and concave down occurs at points of inflection

  • which are located on the midline

• This alternating concavity is directly linked to how the rate of change behaves:

  • When the graph is concave down, the rate of change is decreasing

  • When the graph is concave up, the rate of change is increasing

What symmetry do the sine and cosine functions have?

• The sine function has rotational symmetry about the origin

  • $\sin (-\theta )=-\sin \theta$

    • This makes sine an odd function

  • Graphically, rotating the graph $180°$ about the origin produces the same graph

• The cosine function has reflective symmetry over the $\mathit{y}$-axis

  • $\cos (-\theta )=\cos \theta$

    • This makes cosine an even function

  • Graphically, reflecting the graph across the $y$-axis produces the same graph