Math Is Fun Forum

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Periodic function

Find the period of the following functions
a)f(x) = sin 4x + cos 4x,
b)g(x) = |sin 4x| + |cos 4x|,
c)h(x) = sin 4x + cos 6x.

Thanks in Advance

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Letter, number, arts and science
of living kinds, both are the eyes.

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Periodic function

The first one  , since

The Period is

Last edited by Stanley_Marsh (2007-04-02 04:14:40)

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Numbers are the essence of the Universe

jagabandhu

Member

Registered: 2007-03-30

Posts: 6

Re: Periodic function

see the definition of a periodic function is as below:

A function f(x) is periodic if f(x)=f(x+t) so that t is the period of the function f(x). Now the period of sin(x) and cos(x) can be determined by using the ASTC rule, where A means all, S means Sin, T means Tan and C means Cot positive in 1st, 2nd, 3rd and 4th quadrants. Accordingly the periods of Sin(x) and Cos(x) are 2(pi).

I think you can approach the problem by using this. Otherwise you can ask me again or e-mail me in my id writing the complete problem.

good luck.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

My conjecture is that if m and n are positive integers with greatest common divisor d, sin(mx)±cos(nx) has the same period as sin(dx) or cos(dx), namely 2π⁄d. If so then f(x) and h(x) should have periods 2π⁄4 = π⁄2 and 2π⁄2 = π respectively.

This is just a conjecture though. I‘m not totally sure if it’s right or not (though I’m trying to figure out if it’s true or not). In any case, I’ve plotted the graphs of these particular functions – and they seem to agree with my conjecture.

For the second one, note that

Hence, g(x) = |sin(4x)+cos(4x)| for 0 ≤ x < π⁄8, g(x) = |sin(4x)−cos(4x)| for π⁄8 ≤ x < π⁄4, and the cycle repeats. ∴ its period is π⁄4.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

WAIT! I just plotted the graph of g(x) again and found that its period is actually π⁄8. I must have done something wrong above.

I’ll sort it out and fix it.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

Ahahaha! It’s actually simple enough!

is the exactly the same as

And

Hence the behaviour of g(x) is totally identical for 0 ≤ x < π⁄8 as for π⁄8 ≤ x < π⁄4. ∴ g(x) has a period of π⁄8 after all.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

Now I return to the claim I made earlier: if d = gcd(m,n) then sin(mx)±cos(nx) has period 2π⁄d. This is, of course, provided sin(mx)±cos(nx) ≢ 0. Well, to show that 2π⁄d is a periodicity value, we just replace x by x+2π⁄d and see what happens.

since m⁄d and n⁄d are both integers.

Moreover, since d is the largest integer that makes both m⁄d and n⁄d integers, it is clear that 2π⁄d is indeed the smallest periodicity value for sin(mx)±cos(nx). Well, I suppose that wraps it up.

Note that this also works for sin(mx)±sin(nx) and cos(mx)±cos(nx) (provided they are not identically 0).

Last edited by JaneFairfax (2007-04-03 10:09:59)

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Periodic function

Still fighting the 3rd one , don't think the 3rd is a periodic function

Last edited by Stanley_Marsh (2007-04-03 10:57:45)

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

Yes, it is!

sin{mx}+cos(nx) has period 2π⁄d, where d = gcd(m,n). How many times must I say that?

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Periodic function

Yes, it is!

sin{mx}+cos(nx) has period 2π⁄d, where d = gcd(m,n). How many times must I say that?

How to prove this theorem?

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Periodic function

Now I return to the claim I made earlier: if d = gcd(m,n) then sin(mx)±cos(nx) has period 2π⁄d. This is, of course, provided sin(mx)±cos(nx) ≢ 0. Well, to show that 2π⁄d is a periodicity value, we just replace x by x+2π⁄d and see what happens.

since m⁄d and n⁄d are both integers.

Moreover, since d is the largest integer that makes both m⁄d and n⁄d integers, it is clear that 2π⁄d is indeed the smallest periodicity value for sin(mx)±cos(nx). Well, I suppose that wraps it up.

Note that this also works for sin(mx)±sin(nx) and cos(mx)±cos(nx) (provided they are not identically 0).

Postcount, woohoo.