How to i solve the amplitude 3sin(2x)+4sin(x)

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Discussion Overview

The discussion centers around determining the amplitude of the function 3sin(2x) + 4sin(x). Participants explore different methods for finding the amplitude, including the concept of constructive interference and the use of calculus to find maximum values.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant states that the amplitudes of the individual sine components are 3 and 4, suggesting that they can add constructively to a maximum amplitude of 7 at certain points.
  • Another participant questions the validity of the constructive interference claim, arguing that the two sine functions do not align perfectly to achieve the sum of their amplitudes due to their differing frequencies.
  • A proposed method involves finding the maximum value of the combined function using calculus, leading to an approximate maximum value of 6.10.
  • Participants discuss the use of software for graphing the functions to visualize the components and their sum.

Areas of Agreement / Disagreement

There is disagreement regarding the amplitude calculation, with some participants proposing a maximum amplitude of 7 based on constructive interference, while others argue that the actual maximum is approximately 6.10 based on calculus. No consensus is reached on the correct approach to determining the amplitude.

Contextual Notes

Participants express uncertainty regarding the conditions under which the amplitudes can be added and the assumptions involved in their calculations. The discussion reflects differing interpretations of the interaction between the sine functions.

expscv
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how to i solve the amplitude 3sin(2x)+4sin(x)
 
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The amplitude of one component is 3; the amplitude of the other is 4. When two sine waves are added together, regardless of their frequencies or phases, they will periodically be in phase and add together constructively. They will add to an amplitude of 7 at those points.

- Warren
 
thx , is amplitude find through max/min pts?
 
Far be it from me to disagree with Chroot, but I don't see how sin(x) and sin(2x) will "constructively interfere" to give an amplitude of 3+ 4. 4sin(x)= 4 only for x= pi/2+ 2npi while 3sin(2x)= 3 only for 2x= pi/2+ 2mpi or x= pi/4+ mpi. That means we would have to have (m-2n)pi= pi/4 for some integers m and n.

The way I would do this problem is to find the maximum value of y= 3sin(2x)+ 4sin(x) as expscv suggested:
If y= 3sin(2x)+ 4sin(x) then y'= 6cos(2x)+ 4cos(x). cos(2x)= 2cos2(x)- 1 so this is y'= 12 cos2(x)+ 4cos(x)- 6= 0.

Solving that quadratic equation for cos(x), then converting that to sin(x) and sin(2x) (sin(x)= sqrt(1- cos2(x)) of course, and sin(2x)= 2sin(x)cos(x)), I get that 3sin(2x)+ 4sin(x) has a maximum value of approximately 6.10, the same thing I get by graphing y= 3sin(2x)+ 4sin(x).
 
Whoops. Integer multiple frequencies. :sheepish grin:

- Warren
 
thx both in reply, espically HallsofIvy ,
 
I got too curious. :smile:

The red and blue are the two components. The black one the sum.
 

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cool wat software is that?
 
Nice looking functions. You can do that with Excel as well, although it would be a bit less convenient.
 
  • #10
It's called ROOT. You can download it from here. It is mainly used for high energy physics.
 

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