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  • Perplex

    Complex Numbers (Lesson 4/5)

    Complex Phasors

    1 / 5

    Discussion

    Consider the functions ​f(x)=3sin(x), ​g(x)=5sin(x+2π​). Let ​h(x)=f(x)+g(x).


    Our goal is to write ​h(x)​ in the form ​h(x)=Asin(x+B). Let's walk through how to do this step by step.

    (a)

    First, show that ​Im(rcis(ΞΈ))​ is a sine function.

    Solution:

    We use the definition

    ​
    cis(ΞΈ)=cosΞΈ+isinΞΈ
    ​

    so

    ​
    rcis(ΞΈ)=r(cosΞΈ+isinΞΈ)=rcosΞΈ+irsinΞΈ
    ​

    Taking the imaginary part gives

    ​
    Im(rcis(ΞΈ))=rsinΞΈ
    ​

    which is exactly a sine function of ​θ​ with amplitude ​r.