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 asamplitude = \( |a| \)
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)
Explore how the 4 coefficients a,b,c and d affect the graph of f(x)?
More References Related to Cosine FunctionsProperties of Trigonometric FunctionsGraphs of Basic Trigonometric Functions Unit Circle and Trigonometric Functions sin(x), cos(x), tan(x) |