# Similtanious equations

1. 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?

2. Homework Equations

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

3. 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 ive never done it with three equations before.
i'd appreciate any help :)

Related Calculus and Beyond Homework Help News on Phys.org
Hootenanny
Staff Emeritus
Gold Member
HINT: For what values of the argument is sine zero?

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

Last edited:
Hootenanny
Staff Emeritus
Gold Member
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"?

i understand how you derive that mathematically, but cant 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

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?

how do i find that out with three equations though?

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

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?

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 $\pi$.

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 isnt involved.

i get it. so x=2656.50006565

which is the LCM of the three numbers

right, thanks a lot Hootenanny and Tedjn :)