Find Intersections of Trig Functions with different periods

[vishal--j]

There are 2 trig functions on the same set of axis.

f(x)=600sin(2π3(x−0.25))+1000 and f(x)=600sin(2π7(x))+500

How do I go about finding the points of intersections of the two graphs?

This was from a test I had recently and didn't do too well on,so any help would be much appreciated.
I started by equating the 2 equations but from there I got pretty stuck .

We covered a similar question in class except the values in the brackets in both sides could be set to a variable such as 'y'. Since I can't see a way to do that here I am pretty much stuck.

This is from a final year high school calculus class.

Thank you very much in advance.Sorry I posted this in the maths part of the forum first as I didn't realize there was this part of the forum.

 

[Simon Bridge]

Welcome to PF;
In general - it can get very complicated.
In this case - you start by noticing that the sinusoids have the same amplitude but different wavelengths and starting points.
Start by sketching them out. Notice you can shift both down by 500 and keep the x-intersections the same.
This will help you see where to look.

In an exam you have to use your experience of doing this sort of thing in class so it is worth spending a bit of time understanding what all the bits mean.

 

[vishal--j]

Hey Simon,

Thanks for the welcome. In this test we had to find the equation ourselves from data given in a brief paragraph and I was able to confirm the equations were correct using my graphics calculator. However to get the good grades we had to solve this algebraically (not sure if that is the right term?) using the formulas given in the last page of this document (pdf warning).

I was able to see on the graph that the points of intersection repeated every 21 units on the x-axis but figuring out how to mathematically work these out has just left me feeling confused.

Any help would be very much appreciated.

Thanks

 

[Simon Bridge]

What did you try? - show your working.

 

[HallsofIvy]

Shouldn't "$2\pi3$" and "$2\pi7$" be "$2\pil/3$" and "$2\pi/7$"?

 

[theodoros.mihos]

May your equations are:
$$ f(x) = 600 \sin\left(\frac{2\pi}{3}(x-0.25)\right) + 1000 \, \text{ and } \, g(x) = 600 \sin\left(\frac{2\pi}{7}x\right) + 500 $$
and you ask for the "first" point on what intersects?

 

[Simon Bridge]

The question asks for all the intersection points between f and g - which will be why the talk about periodicity later.
But it's worth checking.

Good catch about the fractions.

 

[vishal--j]

Shouldn't "$2\pi3$" and "$2\pi7$" be "$2\pil/3$" and "$2\pi/7$"?

Yes sorry about that and thanks for pointing that out.

May your equations are:
$$ f(x) = 600 \sin\left(\frac{2\pi}{3}(x-0.25)\right) + 1000 \, \text{ and } \, g(x) = 600 \sin\left(\frac{2\pi}{7}x\right) + 500 $$
and you ask for the "first" point on what intersects?

Yes those are the correct equations. I am actually looking for all points of intersection over an 'extended period' (exact words). This is basically asking for a set of intersections then either a general formula to find these intervals or just listing a few of them, BUT it must be done algebraically.So far I have found out that $\frac{2*pi*x}{7}$ is the same as $\frac{3}{7}(\frac{2*pi*x}{3})$ so my first step would be to substitute this in and all $\frac{2*pi*x}{7}$ could then be replaced by a variable such as 'y' to help simplify.

Thanks everyone for being patient with me it is really great to find a place like this! Sorry about the bad formatting of the equations still getting used to using latex.

EDIT: Will be out at camp for the next couple of days so I won't get to a change to reply.

 

[theodoros.mihos]

Can you see the wavelength for waves represented by these equations? This fact you can drive you to better equations. Take and a graph.