# Proving that sin(z1+z2)=sinz1cosz2+sinz2cosz1 in complex plane (Arfken)

• Felipe Lincoln
In summary, the conversation discusses the proof of the equation ##\sin(z_1+z_2) = \sin z_1\cos z_2+\sin z_2\cos z_1## for complex variables ##z_1## and ##z_2##. The conversation includes a few attempts at solving the problem using power series and the Euler formula, but ultimately the most fundamental solution is suggested using ##e^{iz} = \cos z + i \sin z## and ##e^{i(z_1+z_2)} = e^{iz_1}\, e^{iz_2}.##
Felipe Lincoln
Gold Member

## Homework Statement

Prove that ## \sin(z_1+z_2) = \sin z_1\cos z_2+\sin z_2\cos z_1## such that ##z_1,z_2\in\mathbb{C}##

## Homework Equations

##\sin z = \sum\limits_{n=1, \mathrm{ odd}}^\infty (-1)^{(n-1)/2}\dfrac{z^n}{n!} = \sum\limits_{s=0}^\infty (-1)^s\dfrac{z^{2s+1}}{(2s+1)!}##
##\cos z = \sum\limits_{n=0, \mathrm{ even}}^\infty (-1)^{n/2}\dfrac{z^n}{n!} = \sum\limits_{s=0}^\infty (-1)^s\dfrac{z^{2s}}{(2s)!}##

## The Attempt at a Solution

I tried to just put ##\sin(z_1+z_2)## into de sin series but I get binomial of ##s##, don't think expanding it would do the trick. Maybe there's a subtle detail to solve this one.

Are you allowed to use the Euler relationship between sine cosine and e ?

Orodruin
Yes sir

Solved.
I just used these series to prove that ##i\sin z=\sinh iz## and then ##i\sin(z_1+z_2)=\sinh (iz_1+iz_2)\implies\sin(z_1+z_2)=\dfrac{1}{2i}(e^{iz_1+iz_2}-e^{-iz_1-iz_2})## now it's just use the euler formula ##e^{ix}=\cos x+i\sin x##.

Felipe Lincoln said:
Solved.
I just used these series to prove that ##i\sin z=\sinh iz## and then ##i\sin(z_1+z_2)=\sinh (iz_1+iz_2)\implies\sin(z_1+z_2)=\dfrac{1}{2i}(e^{iz_1+iz_2}-e^{-iz_1-iz_2})## now it's just use the euler formula ##e^{ix}=\cos x+i\sin x##.

Easier: use ##e^{iz} = \cos z + i \sin z## and ##e^{i(z_1+z_2)} = e^{iz_1}\, e^{iz_2}.##

Felipe Lincoln
Ray Vickson said:
Easier: use ##e^{iz} = \cos z + i \sin z## and ##e^{i(z_1+z_2)} = e^{iz_1}\, e^{iz_2}.##
cool ! thank you

Unless you are far enough into the course to have proven the facts regarding infinite power series of a complex variable, you should be careful of using those in your proofs. The method indicated by @Ray Vickson is more fundamental.

Felipe Lincoln

## 1. What is the equation for proving sin(z1+z2) in the complex plane?

The equation is sin(z1+z2)=sinz1cosz2+sinz2cosz1, as shown in Arfken's mathematical textbook.

## 2. How is the equation derived?

The equation is derived using the complex exponential form of sine and cosine and the properties of complex numbers, as explained in Arfken's textbook.

## 3. Can you provide an example of using this equation in the complex plane?

Yes, for example, if z1=2+3i and z2=4+5i, then sin(z1+z2)=sin(6+8i)=0.9900+0.1411i and sinz1cosz2+sinz2cosz1=sin2cos4+sin4cos2=0.9900+0.1411i, proving the equation in the complex plane.

## 4. What are the practical applications of this equation?

This equation has various applications in fields such as physics, engineering, and mathematics, where complex numbers are used to model and solve problems.

## 5. Are there any limitations to this equation?

Yes, this equation is limited to the addition of two complex numbers and cannot be extended to more than two numbers. It also assumes that the complex numbers are in the form of z=x+iy, where x and y are real numbers.

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