# Cosine function

1. Dec 22, 2008

### sara_87

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. Dec 22, 2008

### cristo

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

3. Dec 22, 2008

### sara_87

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

4. Dec 22, 2008

### cristo

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

5. Dec 22, 2008

### HallsofIvy

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

6. Dec 22, 2008

### sara_87

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

7. Dec 22, 2008

### HallsofIvy

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

8. Dec 23, 2008

### sara_87

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

9. Dec 23, 2008

### Staff: Mentor

No. Look at posts 4 and 7.

10. Dec 23, 2008

### HallsofIvy

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

11. Dec 23, 2008

### boombaby

...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. Dec 24, 2008

### HallsofIvy

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

13. Dec 24, 2008

### sara_87

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. Dec 24, 2008

### HallsofIvy

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. Dec 24, 2008

### sara_87

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