I wanted to know, how can I determine the real and imaginary part in $\sin z$ where $z \in \Bbb {C}$? Well, this is a part of a series of questions comprising the same in $\log z$ and $\tan^ {-1...

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There are a number of solutions to this problem online that use identities I have not been taught. Here is where I am in relation to my own coursework: $ \sin (z) = 2 $ $ \exp (iz) - \exp (-iz) = 4i...

\begin {align}&\cos z = \cos\left (x + \mathrm {i} y\right) = \cos x\cosh y - \mathrm {i} \sin x\sinh y& (14)\\\end {align} 这样，就把正余弦的实部和虚部分开来了（当然也可以根据定义直接得到两式）．

Nov 24, 2019 · Theorem Let z = x + iy ∈ C be a complex number, where x, y ∈ R. Let sinz denote the complex sine function. Then: |sinz| = √sin2x + sinh2y where: |z| | z | denotes the modulus of a complex number z z sinx sin x denotes the real sine function sinh …

Feb 9, 2018 · Adding, subtracting and multiplying these two formulae give respectively the two Euler’s formulae. (which sometimes are used to define cosine and sine) and the “fundamental formula of trigonometry ” z = 1. As consequences of the generalized Euler’s formulae one gets easily the addition formulae of sine and cosine:

Dec 8, 2022 · Let z = x + iy ∈C z = x + i y ∈ C be a complex number, where x, y ∈R x, y ∈ R. Let sin z sin z denote the complex sine function. Then: where: cosh cosh denotes the hyperbolic cosine function. From Sine of Complex Number: The result follows by definition of the real part of a complex number. .

How do I find the zeros of $\sin (z)$, where $z$ is a complex number? I know that along the real line we have zeros along $k\pi$, where $k$ is an integer. But what about the rest of the plane? The ...

The sine of a complex number can be found using the formula sin (z) = (e^ (iz) - e^ (-iz))/2i, where z is a complex number and i is the imaginary unit. This formula can be derived using the exponential form of a complex number, z = x + iy = r (cosθ + isinθ), where r is the modulus and θ is the argument.

In programming languages, sin and cos are typically either a built-in function or found within the language's standard math library. For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double).