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# Complex Conjugate of Sin

TopComplex Numbers consist of two parts: Real and Imaginary. It can be purely real or purely imaginary or can include both. A purely real Complex Number has its imaginary part equals to 0, whereas a purely Imaginary Number has its real part equals to 0. With this fact we proceed further on finding Complex Conjugate pairs. General way to represent a conjugate of any complex number is: Conjugate (C + Di) = (C – Di). When we talk about complex conjugate of Trigonometric Functions we Mean taking conjugate of sine, cosine, tangent etc. To find complex conjugate of sin function we need to remember certain things like:
Where, (C – Di) and (C + Di) are called as conjugate pairs. In case of purely imaginary numbers as real part is zero, we have C = 0. So, imaginary complex conjugate pairs can be written as (0 - Di) and (0 + Di). Similarly, while writing real complex conjugates we omit imaginary part as: (C – 0i) and (C + 0i).
Arithmetic operations that we are much familiar with like addition, subtraction, multiplication and division can be applied in same way to complex conjugate pairs also. Let’s understand this with some formulae and examples:
(C + Di) + (C – Di) = 2 C, addition is purely real.
(C + Di) - (C – Di) = 2 Di, subtraction is purely imaginary.
Sine function of Trigonometry can be represented as:
Sin (Z) = 1 / 2 (eiZ – e-iZ),
Where, 'z' is any angle of which we have expanded sine function.
Conjugate of sine function can be written as follows:
Suppose conjugate of Z = Z* = V,
Sin (Z*) = 1 / 2 (eiZ – e-iZ)*,
Sin (V) = 1 / 2 (- eiV + e-iV)*,
Sin (V) = - (1 / 2 (eiV - e-iV)),
Sin (V) = - Sin (Z*).