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.
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?