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In thhis video we want to graph equations of the form r is equal to a
times the sine of (k theta)
or r is equal to a
times the cosine of (k times theta). Where if k
is an odd number
the we get k petals and if k is an even number
we get two k
petals
these two
equations generate curves that are called rose curves
their graphs will resemble flowers with petals attached to them
and in this case where k is an odd number
we get an odd number of petals.
Example:
If r is equal to five
times the sine of (5 theta)
this will have
five petals.
whereas r is equal to three times the cosine
of ( four theta)
will have
8 petals. Another thing to realize is that
these graphs have some symmetry attached to them.
The first curve
will be symmetric
with respect to the line
theta is equal to Pi halves.
and second curve, the one involving cosine,
will be symmetric
with respect to the polar axis.
Another thing to note when graphing these curves,
is you want to find where
is r at a maximum or even a minimum? So for example if r is equal to two times a sine of (three theta).
This will reach a maximum
when
the sine of (three theta) is equal to one. That occurs at three theta
is equal to Pi over 2, or three Pi over two, etc. And r is equal to two
times a sine of ( three theta)
reaches a minimum at r equal to zero and you ask yourself when
is the sine of (three theta)
equal to zero? that is when
three theta
is equal to zero, Pi
2 Pi, etc. Similar arguments can be made
in the case of cosine.
Another question to ask yourself- you want to identify
where the graph is increasing or decreasing?
And that will help you
graph the equation in polar coordinates. So let's graph r is equal to two
sine of (three theta).
Let's recall in rectangular coordinates, if we were just graphing regular rectangular coordinates,
r is equal to two sine of (three x).
wanna do this to try to identify
Where the graph is increasing and decreasing? Where it attains maximums and minimums? And where
we have zeros occuring?
The graph of a sine of (three x).
The period has been shortened. We have a period of Pi.
is our new period.
The two as the amplitude. So up here at 2
and down here and negative 2. We will see the graph
contained within that
that region
and
sine starts fine start out at the origin.
It increases to the maximum and goes back down to zero
it reaches a minimum
gets back to zero
and reaches the maximum, then back down to zero. So if this is Pi
here this would be
Pi over six
and here
is Pi halves.
At this spot here,
this is Pi thirds
and over here at this spot
is two Pi thirds.
and here
is five Pi over six.
And that's one complete the cycle of the graph.
Now in this region, number one
the graph of sine is increasing
from zero to Pi over six,
That is region number one.
in a region number two
between Pi over six
and Pi thirds,
it decreases.
Region number three
it is still decreasing
between
Pi over three
an Pi halves.
And reaches a minimum
value here at Pi halves.
In a region number four, the graph starts to increase again,
between Pi halves
and two Pi thirds.
And still increasing in region number five.
between two Pi thirds
and five Pi over six.
And it reaches the maximum
at five PI over six.
and then in region number six it decreases again, from five Pi over six
to Pi.
so region one, region two, region three
four, region five and region six.
now the equivalent form in polar coordinates,
you will see it will trace out like this.
so these are the coordinate axes.
and here at this
is Pi over six
and only here at
five PI over six
and down here at Pi halves.
so when theta is equal to Pi over six, 5 Pi over six, or Pi halves, that corresponds to features
about our graph.
The graph is attaining a maximum
So if you look at the absolute value of r, is a maximum at each of those locations.
So here is the point
(two
comma Pi over six)
and we have a maximum value of two.
because the sine, the maximum value of sine one times two will give that maximum of 2.
It starts out at the origin.
and
r is increasing
in region number one.
R is increasing to theta equal Pi over six. It attains a maximum at Pi over six.
Now in region number two, the graph is decreasing
back down to zero.
so that
occurs between Pi over six and Pi thirds.
now
in the region number three
the graph is still decreasing
and it attains
the minimum value
at Pi halves.
Region number three is between
Pi thirds
and Pi halves.
and we're starting increase again
from Pi halves
to two Pi thirds.
There is region number four.
And then in region number five,
we are back towards the maximum
to five Pi over six
and finally
we decrease again
from five Pi over six
to Pi, and there is region number six..
You would do a similar argument with cosine,
except you'll see that the cosine will be symmetric about the polar axis.