Graphing sine and cosine curves
We'll begin graphing trigonometric equations with the sine and cosine
functions. They are not only the easiest to graph, but they are the most
useful functions in all of trigonometry. They are used to describe all of the
cyclical mechanisms in nature from the quantum physics level to galaxies,
so you can't go wrong getting a good understanding of how to graph them
and how to write the sinusoidal equations that represent them.
Let's begin with the basics. The first two sinusoidal equations you have
encountered are stripped down to the bone:
I am going to introduce four parameters (A,B,C and D ), not shown in these
equations above, that allow their shapes to be stretched and translated
vertically and horizontally. I'm going to explain how these parameters work
to change the curves' shapes and positions and how to graph the equations
when these parameters are used.
Vertical Translation, also called Vertical Displacement
D is the vertical translation parameter. Think about these two formulas like
this: the functions sin and cos will give you a value depending on what
theta is plugged into them. If you were to plot these functions without a D,
they would look exactly like the two drawings above. However, if you were
to add a D to every value for sin ( theta ) or cos (theta) then every point on
the curves would be moved up or down the amount D.
For example, suppose D is 3. Here is what the two equations would look
like:
and
Likewise, a negative D value would move the curves downward. Suppose D
was -5:
Amplitude
When A is introduced as a coefficient of sin or cos, it multiplied every value
that comes out of those functions by A. Suppose A = 2. Then the functions,
instead of being their normal height of 1, would be 2, and all the points
would be doubled. In other words, the whole function would be stretched
or compressed vertically depending on the value of A. So here is what
happens to a sine curve when it has and amplitude (A) of 2:
The amplitude in a sinusoid curve is always positive. It is the distance from
the midline (D) to the maximum point on the curve.
If the sign infront of the A is negative, that means that the curve is inverted
(upside down)... like this:
The Period
The period of a sine or cosine curve is measured from any point on the
curve on the x axis until the curve begins to repeat itself. Most commonly
that means from zero to 2 pi.
The length of the period in a sine and cosine curve is determined by the
parameter B in this new addition to the equations:
We can use B to calculate the length (period) of the sine or cosine curve.
Assuming that the curve begins at the y axis, we know it will end when B x
is equal to 2 pi. That is because the functions sin and cos ONLY KNOW the
angle values from 0 to 2 pi and then they repeat. So Bx at the end of the
period must be equal to 2 pi.
And if Bx = 2 pi, we can calculate the value of x when the curve reaches the
end of its period to be: x = (2 pi ) / B
Here's a simple example. Suppose B = 2, where does the period end?
Remember that a sinusoid is generated by rotating the terminal side of an
angle starting at the positive y axis, through the first, second, third and
fourth quadrants. Imagine the angle going from the 0 quadrantal, through
the first quadrant, past the pi/s quadrantal, through the second quadrant,
past the pi quadrantal, through the third quadrant, past the 3 pi / 2
quadrantal through the fourth quadrant and then ending at the 2 pi
quadrantal. What we have here is 5 lines (quadrantals) and four spaces
(quadrants) That is not a bad way to view the x points of the sinusoids on
the x axis:
Phase shift
The final parameter we need is one that allows us to translate the curve to
the left or right along the x axis so we can have it begin at any point on the
axis instead of being confined to starting at the y axis. (ie x = 0 )
Here's the new and final formulas containing all of the parameters:
C's job is to move the starting point of the curve along the x axis so we can
start it at any point. In order to know where that x starting value value is,
we must solve for the x value where the curve begins. Since the functions
sine and cosine ONLY KNOW the values between 0 and 2 pi, we can see that
at the beginning of the curve, those functions believe they are at zero.
So we must set what is inside the parentheses to zero and solve for x. That
x will be the phase shift... ie it will be the x value (starting point) when the
entire expression in the parentheses is zero.
This is how the starting x position, known as the phase shift, is calculated
from the parameters in the sinusoidal equations.
Now you have four parameters that can move and reshape the sinusoids.
D, the vertical displacement, moves the curve up and down in the vertical
direction.
A, the amplitude, stretches or shrinks the curve in the vertical direction.
B is used in the formula 2 pi / B to calculate the period.
C is used to find the phase shift (x) in the formula x = C/B
Graphing
Now let's actually go through the procedure of graphing an equation. Since
we can now stretch and move the curve horizontally, we will need a way to
calculate where the five quadrantal values will be located. These lines
determine the shape of the curve. Previously, before we started stretching
and moving the curves along the x axis, they all started at zero and ended at
2 pi. So the values of the quadrantals from the circle were used. That is no
longer possible. Now we have to actually figure out where those regular x
values are that divide up the curve properly.
Two final items.
In order not to get a negative period, B must always be positive.
For example, y = sin ( - 2 x + 1 ) will produce a negative period. No good.
There are two symmetrical identities that will resolve this for sine and
cosine by changing the negative B to a positive B. Very good.
cos(A) = cos(-A) and -sin(A) = sin(-A)
In this example, because it is a sine function, a negative sign will have to be
put in front of the sin and all the terms inside the () signs will be reversed.
so y = sin ( - 2 x + 1 ) becomes y = - sin ( 2x -1)
That second equation is the one you would then graph, disgarding the
original.
In the case of the cos, you can change all the signs inside the () and you do
not have to change the sign in front of the cos. for example if you are given
the equation y = cos( - 2 x + 1 ), it becomes y = cos ( 2 x -1 )
Frequency.
One last item. the frequency of a curve is f = 1 / period.