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Sinusoidal Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

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 $f (θ) = \sin θ$
- Both the sine function and the cosine function are sinusoidal functions
The cosine function is a phase shift (horizontal translation) of the sine function
- $\cos θ = \sin (θ + \frac{π}{2})$
- i.e. $\cos θ$ is the same as $\sin θ$ shifted to the left by $\frac{π}{2}$ 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 $f (θ) = \sin θ$ and $g (θ) = \cos θ$ , the period is $2 π$
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 $\frac{1}{2 π}$
The period and frequency are always reciprocals of each other
- $frequency = \frac{1}{period}, period = \frac{1}{frequency}$

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

For the base functions $f (θ) = \sin θ$ and $g (θ) = \cos θ$
- 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:
- $midline : y = \frac{maximum + minimum}{2}$

For the base functions $f (θ) = \sin θ$ and $g (θ) = \cos θ$
- 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)

Graph of a sinusoidal wave with labelled features: maximum, minimum, amplitude, period, and midline. The wave is red and the midline is blue. — **Key features on a sinusoidal graph**

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
- $\sin (- θ) = - \sin θ$
  - 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 $y$ -axis
- $\cos (- θ) = \cos θ$
  - This makes cosine an even function
- Graphically, reflecting the graph across the $y$ -axis produces the same graph

Examiner Tips and Tricks

When an exam question asks you to describe the "behavior" of a sinusoidal function at a particular input value, it is asking about two things:

Whether the function is increasing or decreasing
and whether the rate of change is increasing or decreasing (which is determined by the concavity)

Thinking about where the input value falls relative to the maximum, minimum, and midline of the function will help you answer these questions. A quick sketch of the graph can make this much easier to see.

Worked Example

The average daily temperature, in degrees Fahrenheit, for a certain city can be modeled by the function $T$ given by $T (t) = 18 \cos (\frac{2 π}{365} (t - 200)) + 54$ , where $t$ is the day of the year for $1 \leq t \leq 365$ . Which of the following best describes the behavior of $T (t)$ on day 170?

(A) The average daily temperature is increasing at a decreasing rate.

(B) The average daily temperature is decreasing at a decreasing rate.

(D) The average daily temperature is decreasing at an increasing rate.

Answer:

A question like this would appear on the calculator part of the exam

So start by using your calculator to graph the function

noting especially what is happening at the point where $t = 170$

Graph on a grid with a black curve peaking at (200, 72), and with the point corresponding to t=170 marked in red. Vertical red dashed line also drawn at t=170.

At that point the graph is just to the left of the maximum

so the temperature is increasing

The point is also clearly between the midline (at $T = 54$ ) and the maximum

so the function is concave down at that point
which means the rate of change is decreasing

(A) The average daily temperature is increasing at a decreasing rate

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