I am supposed to prove that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ using complex exponentials:

$$ \begin{align} \sin\theta&=-\frac{1}{2}i(e^{i\theta}-e^{-i\theta})\\ \cos\theta&=\frac{1}{2}(e^{i\theta}+e^{-i\theta}) \end{align} $$

The proof that I have done is very long and messy and essentially I am showing that LHS and RHS are the same thing. I was wondering if there is a neater proof?



Take the imaginary part of $$e^{i(\alpha+\beta)}=e^{i\alpha}e^{i\beta}$$

  • $\begingroup$ Can't get more succinct than that! You are great at saying what needs to be said very succinctly! $\endgroup$ – Namaste Mar 16 '14 at 13:15
  • $\begingroup$ Thanks dear amWhy but If I improve my English I would give longer answers;-) $\endgroup$ – user63181 Mar 16 '14 at 13:18
  • 1
    $\begingroup$ You actually have a pretty "darn" good mastery of English, Sami! Seriously. $\endgroup$ – Namaste Mar 16 '14 at 13:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy