Don't understand this function, s(cosξ + j sinξ) in my textbook

The textbook is explaining the identity cos θ = se^(jξ) = s( cos ξ + j sin ξ ), where s is the magnitude of a complex number representing cos θ and ξ is its angle in the Gauss plane. This is achieved by converting the complex exponential using Euler's formula. Essentially, they are breaking down cos θ into its real and imaginary parts. In summary, the textbook is presenting the identity cos θ = se^(jξ) = s( cos ξ + j sin ξ ) to explain the relationship between the cosine function and complex numbers.
  • #1
FrankJ777
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6
My textbook for Advanced Electomagnetics, by Balinas has this identity.

cos θ = se^(jξ) = s( cos ξ + j sin ξ ).

I have no idea what they are saying. Is there an S funtion I'm not aware of?
I've looked back and forth, and he doesn't seem to explain it's use.

I've inserted a picture of the page, to provide context.
Hope I'm in the right section.

20200922_183034.jpg
 
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  • #2
I assume ##j^2=-1##, right ? Then s is magnitude of complex number ##cos\theta_l## and ##\zeta## is its angle in Gauss plane.
 
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  • #3
s is just a constant, and they use Euler's formula to convert the complex exponential.
 
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  • #4
So they are just saying that cos θ can be se^(jξ) = s( cos ξ + j sin ξ ). Where s is any old constant?
They're just decomposing cos θ into a real and imaginary part?
 
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  • #5
FrankJ777 said:
They're just decomposing cos θ into a real and imaginary part?
Yes.
 
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FAQ: Don't understand this function, s(cosξ + j sinξ) in my textbook

What does s(cosξ + j sinξ) represent?

The function s(cosξ + j sinξ) represents a complex number in the form of a trigonometric expression. The real part of the complex number is represented by cosξ and the imaginary part is represented by j sinξ.

How do I simplify s(cosξ + j sinξ)?

To simplify s(cosξ + j sinξ), you can use Euler's formula which states that e^(jξ) = cosξ + j sinξ. This will allow you to rewrite the expression as s(e^(jξ)), which may be easier to work with depending on the context of the problem.

What is the significance of the j in s(cosξ + j sinξ)?

The j in s(cosξ + j sinξ) represents the imaginary unit, which is defined as the square root of -1. It is commonly used in mathematics and engineering to represent complex numbers.

Can I graph s(cosξ + j sinξ)?

Yes, you can graph s(cosξ + j sinξ) by plotting the real and imaginary parts of the complex number on the x and y axes, respectively. The resulting point on the graph will represent the complex number in the form of (cosξ, sinξ).

How is s(cosξ + j sinξ) used in science?

s(cosξ + j sinξ) is commonly used in science to represent physical quantities that have both magnitude and direction. It is also used in various mathematical models and equations to solve complex problems in fields such as physics, engineering, and economics.

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