## anonymous 4 years ago Let f(x) = sin(x) + icos(x). Show that |f(x)| = 1. I'm not even sure where to start here...

1. anonymous

hey its too much easy... |f(x)|=sqrt((1)^2+(1)^2)=sqrt(2)

2. anonymous

here we will take its real part 1 will cos(x) and also 1 from imagnary part i(sinx) hey u have written wrong equation.. its cos(x)-isin(x)

3. anonymous

The question is right. I know DeMoivres, but in this case it's sin(x) + icos(x).

4. anonymous

ah oookkk ya its absolute value

5. anonymous

So far I have LHS = $\frac{ e ^{ix}-e^{-ix} }{ 2 } + \frac{ e ^{ix}+e^{-ix} }{ 2 }i$ Have I completely overthought the problem ? :)

6. anonymous

no i is denominator from demorvies, but in this case I didnt get u how u got this result

7. anonymous

ok am gonna solve its prove and implement according to this condition the ll msg u here ok

8. anonymous

Thanks :) I'm so confused...

9. anonymous

hahahhah i just wanna say write i in denominator ok i mean (eix−e−ix)2+(eix+e−ix)2i

10. anonymous

Hmm... I don't get it. How does that show that |sin(x) + icos(x)| = 1?

11. anonymous

oh i see wait i solve it and post the diagrame okk buddy

12. anonymous

Anyone?

13. anonymous

daer i cant .. i have seen the same result for it sqareroot 2 iz answer for both condition.. cos when we see cosx+isinx,, or sinx+icosx... taking coifficients we get the same result..

14. anonymous

i thinks its wrong question or some thing missing ... can u give the link where did u got it from ???

15. anonymous

Yep, it's question 1 iv) from this paper: http://www.scribd.com/doc/112148611/MATH1131-Mathematics-1A-1 Thanks :)

16. anonymous

pdf attached here.

17. anonymous
18. anonymous

u got ur answer or not /??

19. anonymous

Nope, heh. Not yet :(

20. anonymous

ok if its difficault then msg me ur question i will solve it my self and will post the pic here ok

21. anonymous

Yeah I don't know how to get the function to = 1...

22. AccessDenied

I am thinking that you are using the fact that |f(x)| represents the exact distance between it and 0 on the complex plane... Notice that you can create a right triangle with the real part of f(x) and the imaginary part of x on a complex plane: |dw:1352104120632:dw| So, here, Re(f) = sin(x) and Im(f) = cos(x). We drop off the i since we're dealing with distances only for the imaginary part. Using Pythagorean theorem, |f(x)|^2 = (Re(f))^2 + (Im(f))^2 = sin^2 x + cos^2 x. Since we have a neat trigonometric identity here, we can use it to reduce sin^2 x + cos^2 x as 1. Take the square root of both sides and we're done. :)

23. anonymous

Thanks so much for your clear and concise answer!! I've been stuck on this question for a long time :)

24. AccessDenied

You're welcome! :)

25. sirm3d

perhaps this definition can refresh you $\huge \left| a + bi \right|=a^2+b^2$