Go through your notes and tell us the definition of $\cos(x+iy)$. While you're at it check if the plus in $\cos x \cosh y \color{red}+ i\sin x \sinh y$ is correct. $\endgroup$ – Git Gud

2011年5月6日 · One way is to use the power series for sin (x) and cos (x), which are convergent for all real and complex numbers. An easier procedure, however, is to use the identities from the previous section: Any complex number z can be written z = x+ i y for real x and y. We then have. These functions can take any real or complex value, however large.

$$ = e^{iy}\sin(x) + \frac{-e^{-x}e^{iy}-e^{-x}e^{-iy}}{2i}$$ At this point I could have $\displaystyle\frac{-e^{-x}}{i}cos(iy)$ in the right term but I want to make $\cos(x)$ appear, so I don't know how to continue.

2016年4月15日 · How to represent imaginary and real part of $\cos(z)$ as functions of $x$ and $ y$?

2024年3月6日 · In this article, we will see how to calculate the sine, cosine and tangent of a complex variable z. In the process, we will discover how the trigonometry functions sin and cos are related to the hyperbolic functions sinh and cosh. To do that, we need to understand what it means to find the sine of a complex number.

Prove for \cos (x+iy) https://math.stackexchange.com/questions/596694/prove-for-cos-xiy Assuming that you define the cosine on the complex plane by \cos z=\frac{\exp(iz)+\exp(-iz)}{2} you have that \cos is a holomorphic function, which coincides with the cosine defined on the ...

对于复数x+iy，实部x和虚部y都要求是一个单纯的实数，而sin(z)是一个复数，因此不能说它是某个复数的虚部。 相关的公式 ① sin(iy) = i sinh(y) ② cos(iy) = cosh(y) ③ sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y) ④ cos(x+iy) = cos(x)cosh(y) - i sin(x)sinh(y) *备注：cosh(y)和sinh(y)是双曲三角 ...

Using the Euler formula eiy = cosy +isiny, the real sine and cosine functions can be expressed in terms of eiy and e−iy as follows: siny = eiy − e−iy 2i and cosy = eiy + e−iy 2. We define the complex sine and cosine functions in the same manner sinz = eiz − e−iz 2i and cosz = eiz + e−iz 2. The other complex trigonometric ...

\[(\cos \theta + i\sin \theta)^n = \cos n\theta + i\sin n\theta \] 辅助角公式 \[a\sin x + b\cos x = \sqrt{a^2+b^2}\sin (x+\varphi) \] \[\text{其中}\;\varphi\;\text{满足}\cos \varphi = \frac{a}{\sqrt{a^2+b^2}}\,,\sin \varphi = \frac{b}{\sqrt{a^2+b^2}} \]

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any suggestions would be appreciated.