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## #1 2005-09-13 22:41:19

miran
Guest

### extreem of function

hello

I have a function:

1/4(2+Cos[2(a-b)]+Cos[2(a+b)]-2Cos[c]Sin[a]Sin)

I need to know what the a and b should be that this function c would be maximum.

## #2 2005-09-14 02:46:51

ryos
Member
Registered: 2005-08-04
Posts: 394

### Re: extreem of function

The cosine and sine functions are periodic, so they have infinitely many maximums. The period of both is pi radians (180 degrees), though cos begins pi/2 radians to the left of sin. So, at every increment of pi/2, either sin or cos has a maximum.

You can use this info to figure out the answer...

El que pega primero pega dos veces.

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## #3 2005-09-14 08:47:13

Atled
Member
Registered: 2005-08-22
Posts: 9

### Re: extreem of function

1/4(2+Cos[2(a-b)]+Cos[2(a+b)]-2Cos[c]Sin[a]Sin)  <- something seems wrong here

Sin needs a variable.

if a = 0 b= pi/4

you get 1/2

but you might be able to get something smaller than that.

You might try taking partial derivatives of  Cos[2(a-b)]+Cos[2(a+b)] wrt a & b and look for extrema. You could also try to write an excel spread sheet which tries different combinations of a & b and use the goal seek function.

I am in a hurry but I will try to work on it later.

Last edited by Atled (2005-09-14 08:50:32)

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## #4 2005-09-14 09:26:18

MathsIsFun
Registered: 2005-01-21
Posts: 7,660

### Re: extreem of function

miran wrote:

1/4(2+Cos[2(a-b)]+Cos[2(a+b)]-2Cos[c]Sin[a]Sin)

I need to know what the a and b should be that this function c would be maximum.

If you want c to be a maximum, and c is in the function, then we are missing something ... perhaps this function is equal to a constant?

Also if it is cos(c), then c could take on many values, as ryos mentioned already. For example (using radians) cos(1)=0.540, and cos(7.28)=0.540, etc, each time adding 2*pi.

You can also use Excel to plot the function to see how it behaves, and also try different a and b values. This may not get you an exact answer, but could help to point the way.

May I ask what this is related to?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #5 2005-09-14 18:36:08

miran
Guest

### Re: extreem of function

At first i woul like to thank you for your help.

I apologize the corect function is:

1/4(2+Cos[2(a-b)]+Cos[2(a+b)]-2Cos[c]Sin[2a]Sin[2b])

i have simulated this function in program Mathematica and the right solutions seems to be a=b and a=-b or -a=b. Thats experimentaly, but i don´t know how to prove it matematicaly.?

It`s conected with my diploma research work in optical sensors.

## #6 2005-09-14 19:02:17

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: extreem of function

I'm very interested in knowing what c stands for?

igloo myrtilles fourmis

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## #7 2005-09-14 21:22:02

miran
Member
Registered: 2005-09-14
Posts: 1

### Re: extreem of function

I don´t know how to explane it. When i put in Mathematica a and b 45deg (a=b=45deg)
the function is plot (on x axis there´s c and on y axis amplitude) periodical from 0 to 1
in amplitude. Any other combination of a and b for example a=b=20deg will also achive
maximum at 1 but minimum would be in 0.6. The same is if the anyone of a or b is negative.
All this results are good. But if the a and b are not equal the ploted function would not have maximum at 1 but less then 1 and this solution is no good.

The question is how can i calculate from the 1/4(2+Cos[2(a-b)]+Cos[2(a+b)]-2Cos[c]Sin[2a]Sin[2b] a and b teoreticaly because upper results are only practical.

The real solution must be quide easy but i don´t see it.

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