# Quod Erat Demonstrandum

## 2008/04/21

### Complex trigonometric functions

Filed under: University Mathematics — johnmayhk @ 1:03 下午
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F.5 student Hoover found it interesting to know some properties of complex trigonometric functions. He asked if the following is still true?

$\sin^2(z) + \cos^2(z) = 1$

for any complex number $z$.

The answer is affirmative. Here is a verification.

$e^{iz} = \cos(z) + i\sin(z)$

Replace $z$ by $-z$, yields

$e^{-iz} = \cos(z) - i\sin(z)$

By elimination from the equations above, we have

$\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$
$\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$

Now, it is a piece of cake to obtain

$\sin^2(z) + \cos^2(z) = (\frac{e^{iz} + e^{-iz}}{2})^2 + (\frac{e^{iz} - e^{-iz}}{2i})^2 = 1$

Could you try to verify another usual properties of real trigonometric functions to see if they are still applicable to the cases in complex numbers? Like say

$\cos(2z) = \cos^2(z) - \sin^2(z)$

*$\sin(-z) = -\sin(z)$ may be verified by the power series of the sine function.

## 5 則迴響 »

1. 睇過d書講imaginary number既出現係因為處理cubic equation

所以我有個問題想問下John Sir,complex number在實際應用的層面有幾大?

我只知道在solve differential equation等的applied maths會用到,另外亦都聽過d係讀engine,當處理某些問題也會用到complex number

迴響 由 Justin — 2008/04/21 @ 1:42 下午 | 回應

2. 讀 civil engineering 時，其中一科是 electronics。教『三相供電』時，就是把電壓或電流以 complex numbers 表示。相信是為了方便計算而已（即不用 complex numbers 也可）。確實，在工程科目中，complex numbers 是非常有用的。看看這個簡介：

http://en.wikipedia.org/wiki/Complex_number#Applications

迴響 由 johnmayhk — 2008/04/21 @ 3:13 下午 | 回應

3. Here is another nice fact, suppose f, and g are analytic functions in D (here D is a simply connected domain, for example, we may take D to be the open unit disc or the entire complex plane)

Suppose f^2 + g^2 = 1 for z in D.

Then we have f = cos(ih(z)), g = sin(-ih(z)) for some analytic function h in D.

Here is a short proof, we have (f + ig)(f – ig) = 1, and hence f + ig never vanishes in D, thus f + ig = e^{h(z)} for some analytic function h in D. Likewise, f – ig = 1/(f + ig) = e^{-h(z)}.
Therefore, f = [e^{h(z)} + e^{-h(z)}]/2 = cos(ih(z)), and g = sin(-ih(z)). Done.

Hence, sin and cos are essentially the only analytic functions that satisfy f^2 + g^2 = 1.

迴響 由 koopa — 2008/04/22 @ 1:20 下午 | 回應

4. Hi, this is the first time I visit your site, and it’s really nice! Just to share another idea on how to see that $sin^2(z) + cos^2(z) = 1$ holds even for complex numbers: one just needs to observe that both sides of the equation are analytic functions on the entire complex plane, and that the identity holds for real numbers $z$. Then apply the identity theorem for analytic functions. Almost all common trigonometric identities extend to complex arguments this way.

迴響 由 Polam — 2008/04/25 @ 6:54 上午 | 回應

5. Thank you very much for the nice facts given by Koopa and Polam! Yes, the identity theorem should be taught at the very beginning of the complex analysis. I just overlook that. Thank you for reminding me that complex analysis is fun.

迴響 由 johnmayhk — 2008/04/25 @ 3:32 下午 | 回應