Find the Shared y=0 Value for Three Simultaneous Equations | Homework Example

[turnip]

Homework Statement

Three equations (as seen in the relevant equations section) have a y value of 0 when they initially begin at x=0. If x is measured in complete days, how many days will it be before these three functions once again share the y=0 value?

Homework Equations

y = sin (2piex/23)
y = sin (2piex/28)
y = sin (2piex/33)

The Attempt at a Solution

well these equations must all equal 0, therefore y = sin (2piex/23) = sin (2piex/28) = sin (2piex/33) = 0

as to how to manipulate that to get the desired answer, iam not too sure since I've never done it with three equations before.
i'd appreciate any help :)

 

[Hootenanny]

HINT: For what values of the argument is sine zero?

 

[turnip]

well if we are talking radians: pie and 2pie (then obviously continuing on with 3pie, 4 pie,etc)

not quite seeing what your getting at though
unless y = pie or something
and thanks for replying :)

 

[Hootenanny]

well if we are talking radians: pie and 2pie (then obviously continuing on with 3pie, 4 pie,etc)

Correct. So from,

$$\sin\left(\frac{2\pi x}{23}\right) = \sin\left(\frac{2\pi x}{28}\right) = \sin\left(\frac{2\pi x}{33}\right) = 0$$

We can deduce that,

$$\frac{2\pi x }{23} = \frac{2\pi x }{28} = \frac{2\pi x }{33} = n\pi \hspace{2cm}, n\in\mathbb{Z}$$

$$\Rightarrow \frac{2}{23}x = \frac{2}{28}x = \frac{2}{33} x= n \hspace{2cm}, n\in\mathbb{Z}$$

Which in words means that "for which value of x makes all the quotients into integers"?

 

[turnip]

i understand how you derive that mathematically, but can't think as how to solve for it since it says that 2x/23 = 2x/28 which, the way iam looking at it, means that 23 = 28

 

[Tedjn]

What Hootenanny is saying is not really

$$\frac{2\pi x }{23} = \frac{2\pi x }{28} = \frac{2\pi x }{33} = n\pi \text{ where } n\in\mathbb{Z}$$

but

$$\frac{2\pi x}{23} = n_1\pi; \frac{2\pi x}{28} = n_2\pi; \frac{2\pi x}{33} = n_3\pi \text{ where } n_1, n_2, n_3 \in \mathbb{Z}$$

The integers n₁, n₂, and n₃ will not be equal, except at x = 0 (in which case they all equal 0), but what is important is that they are all integers, because sine of any multiple of $\pi$ will equal 0.

What values of x > 0 will make n₁, n₂, and n₃ all integers? Which is the smallest?

 

[turnip]

how do i find that out with three equations though?

 

[Tedjn]

The equations we gave you are the three equations. Each one has two variables, x and n_i, so you cannot find unique answers. However, it is important to note that n_i is an integer. What values of x give integer values of each n_i? What is the smallest? Try to think it out logically.

 

[turnip]

okay, so 2piex/23 = n1

3.660563691=x for n1=1

4.456338407=x for n2=1

5.252113122=x for n3=1

right, so i am to find the lowest common multiple inorder to find at what value of x is when all values of y=0

i know how to do lowest common multiple, for the long way, but i have a feeling this will go on for a while before the three cross paths. What's the simplist way to work out the LCM in this instance?

 

[Tedjn]

You are right, you do need to find the LCM. It would be difficult if there was a $\pi$ in there. Luckily, you wrote down the equations wrong. They are actually

$$\frac{2\pi x}{23} = n_1\pi; \frac{2\pi x}{28} = n_2\pi; \frac{2\pi x}{33} = n_3\pi$$​

Look at what happens to $\pi$.

 

[turnip]

right, thanks for pulling me up on that. so

2/23 = 0.086956521 x=11.5000001
2/28 = 0.071428571 x= 14.00000008
2/33 = 0.06060606 x= 16.50000017

so now iam to do the tedious job of multiplying them out until i find one where all the x's equal the same. is there any easier way of doing that? because this is going to go on for a while even now pie isn't involved.

 

[turnip]

i get it. so x=2656.50006565

which is the LCM of the three numbers


right, thanks a lot Hootenanny and Tedjn :)
and thanks for your patients