Processing math: 100%

Cosine Function

Definition and Graph of the cosine Function

Angle $\theta$ is an angle in standard position with initial side on the positive x axis and terminal side on OM as shown below.

angle in standard position.
The cosine function $\cos(\theta)$ is defined by
$\cos(\theta) = \dfrac{x}{r}$
where $r \$ is the distance of OM where O is the origin of the rectangular system of coordinate and M is any point on the terminal side of angle $\theta$ and is given by
$r = \sqrt{x^2+y^2}$

If point M on the terminal side of angle $\theta$ is such that OM = r = 1, we may use a circle with radius equal to 1 called unit circle to evaluate the sine function as follows:
$cos(\theta) = x / r = x / 1 = x$ : $\cos(\theta)$ is equal to the x coordinate of a point on the terminal side of an angle in standard position located on the unit circle.

No calculator is needed to find $\cos(\theta)$ for the quadrantal angles: $0, \dfrac{\pi}{2}, \pi, ...$ as shown in the unit circle below:
The coordinates of the point on the unit circle corresponding to $\theta = 0$ are: (1,0). The x coordinate is equal to 1, hence $\cos(0) = 1$
The coordinates of the point on the unit circle corresponding to $\theta = \dfrac{\pi}{2}$ on the unit circle are: (0,1). The x coordinate is equal to 0, hence $\cos(\dfrac{\pi}{2}) = 0$
and so on.

unit circle.
Let us now put the values of the quadrantal angles angles $0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2} , 2\pi$ and the values of their cosine on a table as shown below.

$\theta$	$\cos(\theta)$
$0$	$1$
$\dfrac{\pi}{2}$	$0$
$\pi$	$-1$
$\dfrac{3\pi}{2}$	$0$
$2\pi$	$1$

We now plot the points in the above table in a system of rectangular axes

$(x,y)$ and approximate the graph of the cosine function as shown below.

NOTE that we are used to

$x$ being the variable of a function,

$x$ on the graph takes values of

$\theta$ and y takes the values of

$\cos(\theta)$ which is noted as

$y = \cos(x)$ .
After

$2\pi$ , the values of

$\cos(\theta)$ will repeat at the coterminal angles. We say that the cosine function has a period of

$2\pi$ shown below in red.

graph of cos(x) in a rectangular system of axes.

General Cosine Function

We now explore interactively the general cosine function

$f(x) = a \cos(b x + c) + d$

and its properties such as
amplitude =

$|a|$
period =

$\dfrac{2\pi}{|b|}$
phase shift =

$-\dfrac{c}{b}$
by changing the parameters

$a, b, c$ and

$d$ .
Particular exploration of the phase shift is presented by plotting

$f(x) = a \cos(bx + c) + d$ in blue

and

$f(x) = a \cos(bx) + d$ in red (c = 0 and no phase shift)

as shown in the figure below.

cosine function with and without phase shift

You may also want to consider another tutorial on the trigonometric unit circle .
Once you finish the present tutorial, you may want to go through a self test on trigonometric graphs .

Interactive Tutorial on the General Cosine Function

$f(x) = a \cos(bx + c) + d$ in blue

$f(x) = a \cos(bx) + d$ in red (c = 0 and no phase shift)

Press the button "draw" to start graphing cosine functions.

Explore how the 4 coefficients a,b,c and d affect the graph of f(x)?

Amplitude
Set a = 1, b = 1, c = 0 and d = 0. Write down $f(x)$ and take note of the amplitude, period and phase shift (defined above) of f(x).
Now change a , how does it affect the graph?
Period
Set a = 1, c = 0, d = 0 and change b. Find the period from the graph and compare it to $\dfrac{2\pi}{|b|}$ . How does $b$ affect the period of f(x)?
Phase Shift
set a = 1, b = 1, d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to $- c / b$ .
set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to $- c / b$ .
repeat 3 and 4 above for b = 2, 3 and 4.
Vertical Shift
set a, b and c to non zero values and change d. What is the direction of the shift of the graph when d is positive and when d is negative?

More References Related to Cosine Functions

Properties of Trigonometric Functions
Graphs of Basic Trigonometric Functions
Unit Circle and Trigonometric Functions sin(x), cos(x), tan(x)

Cosine Function

Definition and Graph of the cosine Function

General Cosine Function

Interactive Tutorial on the General Cosine Function

Amplitude

Period

Phase Shift

Vertical Shift

More References Related to Cosine Functions