5-4: Composition of Ordinates and Harmonic Analysis

Well, after our long break, we get the strangest lesson that we have had so far...
I will try to help explain things a little bit better.

Basically, this lesson is about graphs that have sin/cos with another sin/cos, and what we have to do is to find the equation. There are two different types (which look different), which are either the sums of the two (cos/sin) or the products.

So...the equations' graphs look like this.
Yes, they are in degree mode ^_^.

Sum :
Blue - y = 3cos (x) + sin (4x)
Orange - y = 3cos (x)
Red - y = sin (4x)















Product :
Blue - y = 3cos(x)sin(4x)
Orange - y = 3cos (x)
Red - y = sin (4x)














Here are some steps to find the equation that were written down during class if you didn't see them.
Well, obviously the first step that you take is to find whether it is a sum equation or a product equation.

If it is the sum of two sinusoids, then...

1) Draw along the sinusoidal axis and determine the equation of the sinusoid you just drew. Usually, if I were to do this, it'd be the larger sinusoid (one with the larger period.)

2) Count the number of mini-cycles within one cycle of the sum of the sinusoids. (Note: this is to help find the one with the smaller period.

3) Determine the Amplitude of the mini-cycle (count the spaces between the larger sinusoid graph and the final graph to find the amplitude)

4) Determine if the cycle is sin or cos

5) Multiply the # of minicylces by the coefficient of x of the sinusoid found in #1.



If the graph is a product of two sinusoids then...

1) Draw the envelope graph.

2) Determine the equation for the envelope graph (the graph with the larger period.)

3) Count the number of mini-cycles within the envelope (the book has a great example of this, Pg 193)

4) Take this number and multiply it by the coefficient of x from #2.

5) Determine if the mini cycles given are sin or cos.
If you cut the envelope in 1/2, then if you

1) Have symmetry, then both trig functions are the same (sinxsinx, cosxcosx.)
2) Don't have symmetry, then both trig functions are different (cos(x)sin(x), sin(x)cos(x) ).

6) Write your product equation.

7) Check on your grapher that you did correctly. (Mr. French did mention that these problems would most likely show up on the calculator portion =].)


Alright, now it's time for YOU to show your proficiency at these types of problems! Hurray?



Find the equation from the graph provided above.

















Method :

First we figure out what type of equation it is, product or sum.
Obviously, it is a product equation because the amplitude is not always equal.

Next we draw the envelope graph, it should look something like this. The Blue line is what you should've drawn.
The envelope graph should be y = 2sin(x).


















Alright, next we count the number of mini-cycles within the envelope, which is 10.
Then we multiply it by the coefficient of x in the larger-period sinusoid, which is 1. So it is still 10.

Now, we have to determine if the other sinusoid is a sin or cos graph. Well, to figure that out, we cut out 1/2 of the envelope and decide what it is. It isn't symmetric, therfore they are different functions. That makes it a cosine function.
From this information, we can now find the other function which is cos10x.


Therefore the answer is...

y = 2sin(x)cos(10x)

Amira...YOU ARE UP NEXT!

Ok...now for my personalization.
For those of you guys who have not yet realized what one of the coolest shows on TV is...
It is no other than Heroes on NBC!!!
It is on tonight at 9 P.M. and watch it if you have time, and if you do happen to miss it, you can always then go onto nbc.com and then watch it there. Hope you guys now understand the lesson now, than before!!