# 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.

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.

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.

## 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.

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.

 a = 1 b = 1 c = 1.5 d = 0
>

Explore how the 4 coefficients a,b,c and d affect the graph of f(x)?
1. ### 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?
2. ### 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)?
3. ### 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 $- c / b$.
4. 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 $- c / b$.
5. repeat 3 and 4 above for b = 2, 3 and 4.
6. ### 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)