Transcript Sine and Cosine - mathdoctor1999.com

Sine and Cosine
Exploring what periodic changes do to their graphs
The BIG Question
Did you prepare for today? Write yes or no on
today’s date on your celeration chart.
If so, estimate the amount of time and
write it on your celeration chart.
If we start with
covered the periods of sine and
y =We
sinhave
(x),already
it
cosine.
takes
the graph
a cycle of 2π to
Who to
recalls their period?
get back
where it started They both have a period of 2π.
at y = 0.
Who recalls what that means?
The period means to make one complete cycle of the graph. In
other words, the period tells us how often the graph goes one
complete repetition around the unit circle or on the interval 0 to
2π . That is, the period is how long it takes for the graph to
return to the same place it started on the interval 0 to 2π.
Cosine Review
Here is y = cos(x) and again it takes a cycle of
2π to get back to where it started at y = 1.
What do you think happens when we mess
with the inside of the sine or cosine function?
If we take the y = sin (x) and y= cos (x) and change
them to: y=sin(Bx) and y=cos(Bx).
That is, what will multiplying x by a real number
B do to the graph of the functions?
Quick Algebra Review
Do you recall what happens in algebra to the
graph of the quadratic function when we
multiplied its equation by a real number?
If we multiplied by a number greater than one, it made
the graph grow faster or taller and skinnier.
If we multiplied by a proper fraction, it slowed the
growth of the graph down or made it wider.
What is a proper fraction?
It is a fraction in which the numerator is smaller
than the denominator.
y = 2x²
y = x²
y = x²/4
y=sin(Bx) and y=cos(Bx)
Do you think that multiplying the sine and cosine
functions by a real number, B, on the inside will have
we are altering the
a similar effect? Yes, because
y = sin (x/2) so
y = sin (2x) so
function’s period
on asthe interval
0 as
cycles half
How?
cycles twice
fast.
tothan
2π.one, B > 1, this will make the function
fast.
By
multiplying
by
a
number
bigger
Multiplying it by a proper fraction, 0 < B < 1 this will make it cycle slower or
give itfaster
a horizontal
cycle
or give itstretch.
a horizontal shrinking.
y = sin (2x)
y = sin(x/2)
y = sin(x)
2π
y = sin (x)
What does horizontal shrinking imply with
respect to cycling or period?
The period will be less than 2π so the function will repeat faster,
like the affect we saw when we multiplied by a number bigger than
one to the quadratic function.
Cycles twice
as fast
Grows twice
as fast.
What does horizontal stretching imply with
respect to cycling or period?
The period will be greater than 2π so the function will repeat
slower, like the affect we saw when we multiplied by a proper
fraction to the quadratic function.
Grows
slower
Cycles
slower
Effect on Period
How does this affect the period?
It changes it to: period =
In the equations we looked at previously, y = sin(x)
and y = cos(x), what is B? 1
Example 1
Discuss the period of y = sin (2x).
Solution: Since B = 2,
•This means that the graph completes a cycle at a period of π and
at 2π it will have completed two cycles, or B many cycles.
•Thus, B tells us how many completed cycles the graph will
make within the interval from 0 to 2π. Here it will be two.
•The graph of the equation y = sin (2x) will have a horizontal
shrink in which it completes one cycle at π and another at 2π.
IMPORTANT question/inquiry/analysis: Why is 2π
in the period divided by B and not multiplied by B?
If we
didproblem,
not divide
in the period,
we
multiply
xfunction
by
equations
**The
period
for we
yby=Bdivide
sin(x)
y =Hence,
cos
(x)
0 B>1
≤ x iny≤the
then
To fix
this
2πand
by when
B,
theis
=2π,
sin(2x)
will
y= sin(Bx)
and y=
cos(Bx),
thesecond
graph ofone
the sine/cosine
function
would
not cycle
complete
one
cycle
at
π
and
a
at
2π.
Thus
the
function
then
the period
for
y
=sin(Bx)
and
y
=cos(Bx)
is
0
≤
Bx
≤
2π.
Thus
faster. In fact, it would cycle slower contradicting what we have seen happen
to has a
horizontal
shrinking.
graphs
ofparts
functions
they
are multiplied
a number
greater
dividing
all
bywhen
B, in
which
B > 0bywe
have,
0 ≤than
x ≤one.
2π/B.
But
there A
is fractional
a graphicalpie
way
to seefaster
this asthan
well.a whole pie!!!!
MORAL:
cycles
Example: y = sin(2x).
Here we would know that the 2
tells us that the sine function
should cycle twice as fast as the
parent sine function.
You try: You try a similar analysis for the value of B
when it is a proper fraction. Can you figure out why
BUT if we multiplied the
when
divide
period ofyou
2π by 2,
then the by a proper fraction that gives the
period would be 4π. This
graph
a horizontal stretch?
tells me the graph would
2π half
a cycle
done
complete one cycle at 4π
NOT 2 cycles by 2π, but half
a cycle at 2π, giving it a
horizontal stretch.
4π one
cycle
done.
Thus it is
cycling
SLOWER!
Example 2
Discuss the period of y = cos(x/2).
Solution: Since B = ½,
How did we
simplify to get
4π?
•This means that the graph completes a cycle at a period of 4π.
•Moreover, at 2π the graph will have completed ½ a cycle.
We know this since B = ½ the graph should complete half a cycle at 2π.
•Thus the graph of the equation y = cos(x/2) will have a horizontal stretch
completing one cycle at 4π.
You try
Discuss the period of y = sin(x/6).
Strategy:
1. Identify B. In this example it is B = 1/6, which is a proper fraction.
2. This means the function will cycle slower so will complete
1/6th of the cycle in the interval 0 to 2π.
3. Period is then B = 2π/(1/6) = 12π.
4. This means that the function will complete one cycle at 12π.
5. Thus, the graph of y = sin(x/6) will have a horizontal stretch completing
one cycle at 12 π. Hence, one complete cycle in (0, 12π).
EXAMPLE 3
Which graph goes with the equation y = sin (x/2)?
Think: which graph
has completed B =1/2
cycles on the interval 0
to2π?
This is the only
graph that
completes half of
a cycle by 2π.
You Try
Which graph goes with y = cos(2x)?
Think: which graph
has completed
B = 2 cycles on the
interval 0 to2π?
This is the only graph
that completes two
full cycles.
Example 5
Graph the equation y = cos(10x).
Strategy:
1. Recall what the parent cosine function looks like.
2. Here B = 10, thus this graph should
cycle10 times between 0 and 2π.
3. This tells me that this graph must have a
horizontal shrinking.
4. The period = 2π/B = 2π/10 = π /5 which
implies one cycle is completed at π /5.
Here one cycle
is completed so
this must be
π/5.
Here is the
tenth cycle,
so this is
where 2π
is.
Can You?
Find 2π/5, 3π/5, 4π/5, etc. on the previous
graph?
Second cycle
complete so
here is where
2π/5
Graph y = cos(x) over the previous
graph?
Fourth cycle
complete so this is
where 4π/5
y=cos(2π)
Application:
The piston movement in a piston engine
can be modeled using the sine
function.
Go to the web site:
http://www.intmath.com/trigonometricgraphs/2-graphs-sine-cosine-period.php
This is the
picture at the
web site.
Activity
• Put in different values of B, specifically, B = .2,
.5, 4, 6.
• Play with how different values of B affect the
cycle of the sine function and how it affects
the motion of the piston.
• Write a short summary of the number of
cycles completed by the sine function for each
of the given values and how the piston was
affected for each given value of B.
• What is the interval covered by the graph?
Screen Shot
Here is the screen shot for B = .2.
How many cycles would the sine function complete for this
value of B?
How would this effect the rate of rotation in the piston?
Screen Shot
Here is the screen shot for B = .5.
How many cycles would the sine function complete for this
value of B?
How would this effect the rate of rotation in the piston?
Screen Shot
Here is the screen shot for B = 4.
How many cycles would the sine function complete for this
value of B?
How would this effect the rate of rotation in the piston?
Screen Shot
Here is the screen shot for B = 6.
How many cycles would the sine function complete for this
value of B?
How would this effect the rate of rotation in the piston?
Thank – You for your
attention and
participation!