## anonymous 5 years ago Can anybody point me to online resources that talk about using de moivre's theorem to prove trig identities like below: sin 2x = 2 cos x. sin x

1. watchmath
2. anonymous

hmm. the idea is this. $e^{i\theta}=cos(\theta)+isin(\theta)$ and $(e^i\theta)^2=e^{2i\theta}=(cos(\theta)+isin(\theta))^2=cos({2\theta})+isin(2\theta)$

3. anonymous

square $(cos(\theta) + i sin(\theta))$ to see what you get. then equate the real part to the real part and you get an identity for $cos(2\theta)$ and another one for $sin(2\theta)$

4. anonymous
5. anonymous

when you compute $(cos(\theta)+isin(\theta))^2$ you get $cos^2(\theta)-sin^2(\theta)+i\times 2 cos(\theta)sin(\theta)$ the real part is $cos^2(\theta)-sin^2(\theta)$ so that must equal the real part of $cos(2\theta)+isin(2\theta)$ which is just $cos(2\theta)$ telling you that $cosd(2\theta)=cos^2(\theta)-sin^2(\theta)$

6. anonymous

likewise $sin(2\theta)=2cos(\theta)sin(\theta)$

7. anonymous

My concern is: what de moivre's theorem says? z^n = r^n (( cos nx) + i sin (n x)) or (cos x + i sin x) ^n= ( cos nx) + i sin (n x) I am confused.

8. anonymous

both are true.

9. anonymous

How come> can u explain?

10. anonymous

$z=re^{i\theta}=r(cos(\theta)+isin(\theta))$

11. anonymous

$z^n=r^n(e^{i\theta})^n=r^ne^{ni\theta}$

12. anonymous

that by the laws of exponents.

13. anonymous

and since $e^{ni\theta}=cos(n\theta)+isin(n\theta)$ you get the second equality

14. anonymous

if you have not seen $z=re^{i\theta}$ as a representation of a complex number, then it requires a difffernt explanation, but if you have seen it it is nothing more than the laws of exponents.

15. anonymous

succinctly put here http://en.wikipedia.org/wiki/De_Moivre%27s_formula

16. anonymous

Thanks .I need to site on this, it has been itching my head since yesterday.I appreciate your help.

17. anonymous

welcome hope at least second explanation was clear.

18. anonymous

i must have made another algebra error let me check.

19. anonymous

if you have not seen $z=re^{i\theta}$ as a representation of a complex number, then it requires a difffernt explanation, but if you have seen it it is nothing more than the laws of exponents.