如何改写三角函数到复数

cdstudio

如何将  a*sin(k*t-f) 改写成 a+b*i 这种简单形式, 不要中间存在exp()函数.

FreddyMusic

只有这个，其他我相信都是不正确的或是未经数学证明的。

http://en.wikipedia.org/wiki/Trigonometric_functions
Relationship to exponential function and complex numbers

It can be shown from the series definitions[5] that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:

  This identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by e ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.
Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:

cdstudio

多谢LS, 这种方法我也知道, 但是无法化简成为a+b*i的简单形式, 看看还有没有人知道其它方法.

tianyikillua

...这个？？？

tianyikillua

先写成极坐标式（e），然后再换为直角坐标式（a+bi）

cdstudio

问了一个数学牛人, 得到的答案是不能化成如此简单的形式, 必须要有exp()函数. 估计这个结果已成定局, 再次感谢大家的帮忙