Complex Notation Homework: Solve for B & Phi in Terms of A, Omega, Delta

In summary, you can use Euler's formula to find B and \Phi in terms of A, \omega, and \delta if B is real.
  • #1
w3390
346
0

Homework Statement



If x= Acos([tex]\omega[/tex]t + [tex]\delta[/tex]), then one can also write it as x = Re(B[tex]e^{i\Phi}[/tex]). Find B and [tex]\Phi[/tex] in terms of A, [tex]\omega[/tex], and [tex]\delta[/tex] if B is real.

Homework Equations





The Attempt at a Solution



Not sure where to start on this one. I know you guys can't give answers. All I'm looking for is where to get started. Any help would be appreciated.
 
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  • #2
w3390 said:

Homework Statement



If x= Acos([tex]\omega[/tex]t + [tex]\delta[/tex]), then one can also write it as x = Re(B[tex]e^{i\Phi}[/tex]). Find B and [tex]\Phi[/tex] in terms of A, [tex]\omega[/tex], and [tex]\delta[/tex] if B is real.

Homework Equations





The Attempt at a Solution



Not sure where to start on this one. I know you guys can't give answers. All I'm looking for is where to get started. Any help would be appreciated.

Are you familiar with converting between the rectangular and polar forms of complex numbers? See partway down this wiki page:

http://en.wikipedia.org/wiki/Polar_coordinate_system

.
 
  • #3
Actually, I think I might have something.

Euler's formula says: e^(i*phi) = cos(phi) + i*sin(phi)

The real part of this is: Re(e^(i*phi)) = cos(phi).

Therefore, the real part of Be^(i*phi) is: Bcos(phi).

So I have: X = Bcos(phi) and X = Acos(wt + delta)

Am I able to just compare the two equations to get the following relationships:

B = A

PHI = wt + deltaIt can't be that simple, can it?
 
  • #4
w3390 said:
Actually, I think I might have something.

Euler's formula says: e^(i*phi) = cos(phi) + i*sin(phi)

The real part of this is: Re(e^(i*phi)) = cos(phi).

Therefore, the real part of Be^(i*phi) is: Bcos(phi).

So I have: X = Bcos(phi) and X = Acos(wt + delta)

Am I able to just compare the two equations to get the following relationships:

B = A

PHI = wt + delta


It can't be that simple, can it?

:biggrin:
 

Related to Complex Notation Homework: Solve for B & Phi in Terms of A, Omega, Delta

1. What is complex notation and why is it used in scientific calculations?

Complex notation is a mathematical representation that allows for the simplification and manipulation of complex numbers. It is used in scientific calculations because it provides a more concise and efficient way to perform operations on complex numbers.

2. How do you solve for B and Phi in terms of A, Omega, and Delta?

To solve for B and Phi in terms of A, Omega, and Delta, you can use the following equations:

B = A cos(Phi) and Phi = arctan( (Omega * Delta) / A )

3. Can complex notation be used for any type of mathematical problem?

Yes, complex notation can be used for any type of mathematical problem that involves complex numbers. It is particularly useful in calculations involving electrical engineering, quantum mechanics, and signal processing.

4. What are the benefits of using complex notation in scientific calculations?

Some of the benefits of using complex notation include simplified calculations, easier visualization of complex numbers, and the ability to represent complex numbers in a compact form. It also allows for the use of powerful mathematical tools such as Euler's formula and the complex plane.

5. How can I check if my solution for B and Phi is correct?

You can check your solution for B and Phi by plugging in the values for A, Omega, and Delta into the equations and verifying that they satisfy the original equation. You can also use a calculator or online tool to convert between complex and polar form and compare your solution to the result.

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