Fundamental Period of f(t) = sin(6t) + cos(8t)

[Larrytsai]

Homework Statement

Let f(t) = sin(6t) + cos(8t).
(a) What is the fundamental period of f(t)?

T= 2pi$$/$$$$\omega$$

I know the fund. period of sin is pi/3 and cos is pi/4,

and the definition of fundamental period saids that f(t + T ) = f(t)

for the smallest T in the function, so would pi/4 be my answer?

 

[Filip Larsen]

Do you know the fundamental period of each of the two terms? If so, let's call them T_s and T_c. If you imagine a plot of the two terms (or of f(t)), then if you mark the two times repeatedly along the time axis you must know find the smallest time T where there is both a T_s and a T_c mark. Another hint may be to think of this as a problem of finding a common denominator.

 

[Larrytsai]

hmm k so if i find the common denominator, it would be 12 correct?

 

[Larrytsai]

I understand what you mean, that we want to find a period where we can say the function f(t) is periodic not just each component, so where sin and cos both begin and end, but I can't see how I would find the period mathematically.

 

[Filip Larsen]

The common denominator for 6 and 8 is not 12, but ... ?

 

[Larrytsai]

The common denominator for 6 and 8 is not 12, but ... ?

for 6 and 8 is 1?,

but i thought we would put it in the formula for a period which is T = 2pi/frequency, and doing so i would get pi/3 and pi/4

 

[Filip Larsen]

My mistake for being imprecise with the 6 and 8.

You are quite right that you need to write up the period of the two terms, like T_s = 2pi/6 = pi/3, and I guess when you said 12 you meant T = pi/12.[STRIKE] In that case you are on the right track and just need to convert that period back to a frequency[/STRIKE].

Later: *sigh* I think I better not mix work and homework assistance in the future.

If you have the two periods as pi/3 and pi/4, you need to find the least time T that is an integral number of those two.

 

[Filip Larsen]

I'm a bit unhappy about having provided such confusing help, so I hope the homework is either done by now or that I do not spoil it too much by revealing the general method to calculate this.

Given a set of real functions f_i such that each function has the fundamental period S (for cos and sin, S would be 2pi), the sum of these functions

(1) $$f(t) = \sum_i a_i f_i(S\frac{n_i}{d_i}t + p_i)$$

where n_i and d_i are natural numbers and a_i <> 0 and p_i are arbitrary constants, then has the fundamental period of

(2) $$T = S\; lcm(\frac{n_i}{d_i})$$

where lcm is the least common multiple of all the fractions $n_i/d_i$. To calculate lcm of fractions one can use the method of decomposing them into prime products with negative powers, as described in [1].

In the case given above we have S = 2pi, n₁ = n₂ = 1, d₁ = 6 and d₂ = 8, which gives T = 2pi lcm(1/6,1/8) = 2pi lcm(2^-13^-1,2^-33⁰) = 2pi 2^-13⁰ = 2pi 1/2 = pi. [1] http://en.wikipedia.org/wiki/Least_common_multiple##Fundamental_theorem_of_arithmetic