Transforming Complex Solutions into Polar Form

In summary: So G is real and x is complex.Why isn't x real?it's the product of two complex numbers … it's very unlikely to be realI am confused then because my question was from a test prep sheet from my professor. Should I perhaps only consider the real part of x(t)?dunno :redface:
  • #1
w3390
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Homework Statement



Show that the solution x(t) = Ge^(iwt), where G is in general complex, can be written in the form x(t) = Dcos(wt - [tex]\delta[/tex]).

D(w) and [tex]\delta[/tex](w) are real functions of w.

Homework Equations



z = Ae^(i[tex]\phi[/tex])

The Attempt at a Solution



So I know I should start by writing G in polar form. I am confused though as to how to go to polar form with just the G. Is it simply just Ge^(i[tex]\phi[/tex]). Then, I could use Euler's formula to write:

Ge^(i[tex]\phi[/tex]) = Gcos([tex]\phi[/tex]) + iGsin([tex]\phi[/tex]).

I am not sure where this gets me. Any help on where to go from here or if this is even correct would be much appreciated.
 
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  • #2
hi w3390! :smile:

(have a delta: δ and a rho: ρ and a phi: φ and an omega: ω :wink:)
w3390 said:

Homework Statement



Show that the solution x(t) = Ge^(iwt), where G is in general complex, can be written in the form x(t) = Dcos(wt - [tex]\delta[/tex]).

D(w) and [tex]\delta[/tex](w) are real functions of w.

but that's obviously not true …

the RHS is real, but x isn't :confused:
 
  • #3
Why isn't x real?
 
  • #4
it's the product of two complex numbers … it's very unlikely to be real
 
  • #5
I am confused then because my question was from a test prep sheet from my professor. Should I perhaps only consider the real part of x(t)?
 
  • #6
dunno :redface:

maybe :smile:
 
  • #7
What I'm saying is:

x(t) = Ge^(i[tex]\phi[/tex])

x(t) = G[cos([tex]\omega[/tex]t - [tex]\delta[/tex]) + i*sin([tex]\omega[/tex]t - [tex]\delta[/tex])

Then taking only the real part of this:

x(t) = Gcos([tex]\omega[/tex]t - [tex]\delta[/tex]).

From here, I can compare to the given solution of x(t) = Dcos([tex]\omega[/tex]t - [tex]\delta[/tex]) and say that G = D.

Does this make sense?
 
  • #8
Hi w3390! :smile:

(what happened to that δ φ and ω i gave you? :confused:)

I don't understand where your second line came from …

w3390 said:
x(t) = G[cos([tex]\omega[/tex]t - [tex]\delta[/tex]) + i*sin([tex]\omega[/tex]t - [tex]\delta[/tex])
 
  • #9
Write [itex]G= re^{i\theta}[/itex]. Then [itex]Ge^{i\omega t}= r e^{i(\omega t+ \theta)}= r (cos(\omega t+ \theta)+ i sin(\omega t+ \theta)[/itex].
 

FAQ: Transforming Complex Solutions into Polar Form

1. What is the complex to polar form conversion?

The complex to polar form conversion is a mathematical process that represents a complex number in the form of a magnitude and an angle. It is used to simplify complex numbers and make them easier to work with in calculations.

2. How is the polar form of a complex number written?

The polar form of a complex number is typically written as r(cos θ + i sin θ), where r is the magnitude and θ is the angle in radians. It can also be written as r∠θ, using the polar coordinate notation.

3. What is the relationship between the polar form and rectangular form of a complex number?

The polar form and rectangular form of a complex number are equivalent representations of the same number. The rectangular form is written as a + bi, where a and b are real numbers, while the polar form is written as r(cos θ + i sin θ). The relationship between the two forms is defined by the trigonometric functions cosine and sine.

4. How do you convert a complex number from polar form to rectangular form?

To convert a complex number from polar form to rectangular form, you can use the following formula: a + bi = r(cos θ + i sin θ). The real part (a) is equal to r cos θ and the imaginary part (b) is equal to r sin θ. Simply plug in the values for r and θ to find the rectangular form of the complex number.

5. What are the practical applications of using complex to polar form conversion?

The complex to polar form conversion is commonly used in engineering, physics, and other scientific fields. It is particularly useful in applying complex numbers to problems involving rotations and periodic functions. It is also used in circuit analysis, signal processing, and control systems.

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