Cosine function

1. sara_87

773
1. The problem statement, all variables and given/known data

what is cos(2n*pi)

2. Relevant equations

3. The attempt at a solution

I understand that cos(npi)=(-1)^n
so is cos(2n*pi)=2(-1)^n ??

2. cristo

8,394
Staff Emeritus
Make the substitution m=2n in cos(mπ)=(-1)^m

3. sara_87

773
so you mean cos(2n^2)=(-1)^2n
??

4. cristo

8,394
Staff Emeritus
No, I mean cos(2n pi)=(-1)^{2n}

5. HallsofIvy

40,307
Staff Emeritus
Do you know what cos(0) is? Do you know that cosine is periodic with period 2 pi?

6. sara_87

773
yep cos0=1
yep, cos is periodic with period 2pi

7. HallsofIvy

40,307
Staff Emeritus
So cos(2n pi)= cos(0+ n(2pi))= ?

8. sara_87

773
oh right,
so =(-1)^(n+1)
is that right?

Staff: Mentor

No. Look at posts 4 and 7.

10. HallsofIvy

40,307
Staff Emeritus
Okay, what does "periodic" mean????

11. boombaby

130
...I think it is necessary to know the graph of cos(x), which may help a lot. so, find one.

edit (:shy: trying not to be ambiguous)
...I think it is necessary for one to know the graph of cos(x), which may also help a lot. (regardless of this particular problem)...
"periodic" is really the key

Last edited: Dec 24, 2008
12. HallsofIvy

40,307
Staff Emeritus
It might help. It is not necessary. All that is necessary is to know what "periodic" means. No computation is required.

13. sara_87

773
I understand that periodic means that cosine function repeats after multiples of 2 pi. but how would that have anything to do with writing cos(2n*pi) ?
cos (npi)=(-1)^n because as long as n is an integer, the value will alternate from -1 and 1 (clearly form the graph)

14. HallsofIvy

40,307
Staff Emeritus
Would it be easier if it were written n*(2pi) rather than 2n*pi? This is about multiples of 2pi!

cos(2pi)= cos(0+ 2pi)= cos(0)= 1

cos(4pi)= cos(2pi+ 2pi)= cos(2pi)= 1

cos(6pi)= cos(4pi+ 2pi)= cos(4pi)= 1

15. sara_87

773
oh right!!! so cos(n2pi) has to always be 1...i feel very stupid, i should have known that. for all n, cos(2npi) must be 1 as long as n is an integer.

thank you very much