Yahoo Answers is definitely the place everyone goes to find an answer for a math question. Otherwise, there is no explanation for the kinds of questions you can find there. Today's question is about simplification of a trigonometric expression:
Simplify: (cos 3x)(1 - cos 2x + cos 4x - cos 6x)
To solve that, we will need some major identities: the sum of cosines, the difference of cosines, the sum of sines, and the product of sines and cosines. With these formulas, this question can be easily solved.
Let's start with the formulas:
cos α + cos β = 2cos((α + β)/2)cos((α - β)/2)
cos α - cos β = -2sin((α + β)/2)sin((α - β)/2)
sin α + sin β = 2sin((α + β)/2)cos((α - β)/2)
sinαcosβ = 1/2(sin(α + β)sin(α - β))
Now, to avoid complications, let's aim for the average. We can turn 1 into cos 0, so we have the cosines of 0 and 6x and the cosines of 2x and 4x. In both formulas they are divided by two, so they should probably cancel each other.
cos 0 - cos 6x = -2sin((6x + 0) / 2)sin(6x - 0)/2 = -2sin2 3x
cos 4x - cos 2x = -2sin((4x + 2x)/2)sin((4x - 2x)/2) = -2sin 3x * sinx
In conclusion, the problem has just become:
(cos 3x)(-2sin2 3x - 2sin 3x * sin x)
Distribute:
-2sin 3x * sin 3x * cos 3x - 2sin 3x * cos 3x * sinx
Transform into a sum of a cosine and a sine (I picked α and β to be 3x, because there will be a sine of zero, which is zero):
-2 * 1/2 * (sin(3x + 3x) + sin(3x - 3x)) * sin 3x - 2 * 1/2 * (sin(3x + 3x) + sin(3x - 3x)) * sin x
-sin 6x * sin 3x - sin 6x * sin x
We are getting really close now, just hold on:
-(sin 6x)(sin 3x + sin x)
-(sin 6x)(2sin((3x + x)/2)cos((3x - x)/2)
-2sin 6x * sin 2x * cos x
And that's it.
Nadav
nadavs
Tuesday, July 29, 2008
Cosine Simplification
Subscribe to:
Post Comments (Atom)
7 comments:
Hello !.
might , perhaps very interested to know how one can collect a huge starting capital .
There is no initial capital needed You may start earning with as small sum of money as 20-100 dollars.
AimTrust is what you thought of all the time
AimTrust represents an offshore structure with advanced asset management technologies in production and delivery of pipes for oil and gas.
It is based in Panama with affiliates around the world.
Do you want to become really rich in short time?
That`s your choice That`s what you wish in the long run!
I feel good, I began to take up income with the help of this company,
and I invite you to do the same. It`s all about how to choose a proper companion utilizes your money in a right way - that`s the AimTrust!.
I make 2G daily, and what I started with was a funny sum of 500 bucks!
It`s easy to start , just click this link http://paqyxameh.the-best-free-web-hosting.com/finefawa.html
and lucky you`re! Let`s take this option together to get rid of nastiness of the life
Hello !.
might , perhaps very interested to know how one can manage to receive high yields .
There is no need to invest much at first. You may commense earning with as small sum of money as 20-100 dollars.
AimTrust is what you thought of all the time
The firm represents an offshore structure with advanced asset management technologies in production and delivery of pipes for oil and gas.
Its head office is in Panama with affiliates around the world.
Do you want to become an affluent person?
That`s your chance That`s what you really need!
I`m happy and lucky, I began to take up income with the help of this company,
and I invite you to do the same. It`s all about how to select a proper partner utilizes your money in a right way - that`s AimTrust!.
I take now up to 2G every day, and my first deposit was 1 grand only!
It`s easy to get involved , just click this link http://ynecynujus.virtue.nu/uduleh.html
and go! Let`s take our chance together to become rich
Your formula:
sinαcosβ = 1/2(sin(α + β)sin(α - β))
appears wrong:
I think it's
sinαcosβ = 1/2(sin(α + β)+ sin(α - β))
Good day, sun shines!
There have been times of hardship when I felt unhappy missing knowledge about opportunities of getting high yields on investments. I was a dump and downright stupid person.
I have never thought that there weren't any need in big initial investment.
Now, I'm happy and lucky , I begin take up real income.
It's all about how to select a proper companion who utilizes your money in a right way - that is incorporate it in real deals, parts and divides the profit with me.
You can get interested, if there are such firms? I have to tell the truth, YES, there are. Please get to know about one of them:
http://theinvestblog.com [url=http://theinvestblog.com]Online Investment Blog[/url]
Hi!
You may probably be very curious to know how one can make real money on investments.
There is no need to invest much at first.
You may begin to get income with a money that usually is spent
for daily food, that's 20-100 dollars.
I have been participating in one project for several years,
and I'm ready to share my secrets at my blog.
Please visit blog and send me private message to get the info.
P.S. I make 1000-2000 per day now.
http://theinvestblog.com [url=http://theinvestblog.com]Online Investment Blog[/url]
Yes undoubtedly, in some moments I can say that I approve of with you, but you may be considering other options.
to the article there is still a suspect as you did in the downgrade delivery of this beg www.google.com/ie?as_q=dfx for windows media player 8.36 ?
I noticed the phrase you suffer with not used. Or you partake of the dreary methods of inspiriting of the resource. I possess a week and do necheg
http://techcommstout.net/alumni/node/8448
Post a Comment