Intersection of a circle and a sine curve

[vcsharp2003]

Homework Statement

What is the maximum number of points at which a circle and the graph of $y =\sin{x}$ can intersect?

Relevant Equations

$y =\sin{x}$
$(x-a)^2 + (y-b)^2 = r^2$

The question is asking for the number of different solutions to the following two equations.
$$y=\sin{x}$$
$$(x-a)^2 + (y-b)^2 = r^2$$

Solving these is complex for me due to one of the equations being a trigonometric function. If I substitute y from the first equation into the second equation, I get the following.
$$(x-a)^2 + (\sin{x}-b)^2 = r^2$$
$$x^2-2ax+a^2 + \sin^{2}{x} -2b\sin{x} +b^2 = r^2$$
$$(x^2 + \sin^{2}{x})-2(ax+b\sin{x}) +(a^2 + b^2) = r^2$$

Normally an equation of degree 2 would have at most two different solutions, but even though the above equation is of degree 2 in $x$, the introduction of $\sin{x}$ probably means its not. I am stuck beyond this, so please help.

Can someone help me determine the number of different solutions?

 

[.Scott]

I saw the answer at once without using any Math.
I suggest drawing a picture using different size circles.
If you are still stuck later today, say so and I will pass you another clue.

 

[vcsharp2003]

I saw the answer at once without using any Math.
I suggest drawing a picture using different size circles.
If you are still stuck later today, say so and I will pass you another clue.

I tried your recommendation on the desmos.com graphing calculator and I was always getting 2 points of intersection for various circles that I tried.

Below are three scenarios that I tried, and they all always intersect at two points. But logically, I cannot arrive at this conclusion of two points. Also, I can understand that if the circle has its center very distant above or below the x or y axis and having a small enough radius then such a circle would have no intersection points with the sine curve.

 

[FactChecker]

Spoiler

Now imagine a huge circle where the top is practically a straight line. Shift it down so that the top is practically on top of the X-axis. How many times can it intersect that sin wave?

 

[vcsharp2003]

[SPOILER="Now imagine a huge circle where the top is practically a straight line. Shift it down so that the top is practically on top of the X-axis. How many times can it intersect that sin wave?"
[/SPOILER]

If it's a straight line there would be many peaks of the sin wave that it would touch. If we take it further down so it is on the x-axis then also there would be many points on the sine curve that it would intersect.

 

[FactChecker]

If it's a straight line there would be many peaks of the sin wave that it would touch. If we take it further down so it is on the x-axis then also there would be many points on the sine curve that it would intersect.

Right. Is the problem statement complete and accurate?

 

[vcsharp2003]

I saw the answer at once without using any Math.
I suggest drawing a picture using different size circles.
If you are still stuck later today, say so and I will pass you another clue.

Please give me the clue as I have tried your suggestion, but still I can't see the logic.

 

[vcsharp2003]

Right. Is the problem statement complete and accurate?

Yes, the problem seems accurate. It's a challenge problem in a high school text book.

 

[willem2]

Yes, the problem seems accurate. It's a challenge problem in a high school text book.

You didn't try to move the circle. The problem statement doesn't forbid this.

 

[BvU]

Red circle: $(x-\pi/2)^2+(y-999)^2=1000^2\$ blue sine: $y = \sin x\$

https://www.wolframalpha.com/input?i=(x-pi/2)^2+(y-999)^2=1000^2+&+y+=+sin(x)

$\$

 

[FactChecker]

Over the range of allowable a, b, r, parameters there is no maximum. It might be possible to determine (or bound) the number of intersections as a function of the parameters, but that is a much harder problem.

 

[FactChecker]

View attachment 363148
Red circle: $(x-\pi/2)^2+(y-999)^2=1000^2\$ blue sine: $y = \sin x\$

https://www.wolframalpha.com/input?i=(x-pi/2)^2+(y-999)^2=1000^2+&+y+=+sin(x)

$\$

Good one. A simple version would be $x^2+(y-r)^2 = r^2$, where $r$ can be a parameter. You can easily get as many intersections as you want.

 

[vcsharp2003]

Right. Is the problem statement complete and accurate?

Sorry, you're correct. I didn't paste the original question at the start. The exact problem as it appears in the Math book is as below. From what I have gathered so far from all the replies is that the answer to this exact question is none of the given options since as the curvature of the circle decreases the number of intersections increases.

and

 

[vcsharp2003]

View attachment 363148
Red circle: $(x-\pi/2)^2+(y-999)^2=1000^2\$ blue sine: $y = \sin x\$

https://www.wolframalpha.com/input?i=(x-pi/2)^2+(y-999)^2=1000^2+&+y+=+sin(x)

$\$

Thank you so much for this example. It helped greatly in understanding the problem.
It seems that the number of intersections has no upper limit.

 

[hutchphd]

Sorry, you're correct. I didn't paste the original question at the start. The exact problem as it appears in the Math book is as below. From what I have gathered so far from all the replies is that the answer to this exact question is none of the given options since as the curvature of the circle decreases the number of intersections increases.

View attachment 363154and

Is anyone else appalled by the syntax of this question? Is (e) a correct answer? I cannot decide.
@vcsharp2003 What is the book and who publishes it ? Bring out the pitards!
I am reminded of Prof. Feynman's tale: https://rangevoting.org/FeynTexts.html

 

[FactChecker]

Is anyone else appalled by the syntax of this question? Is (e) a correct answer? I cannot decide.
@vcsharp2003 What is the book and who publishes it ? Bring out the pitards!
I am reminded of Prof. Feynman's tale: https://rangevoting.org/FeynTexts.html

Until you mentioned it, I didn't notice that it was ambiguous. I took it as "given all possible circles, what is the maximum number of times any circle can intersect the sine function." So answer e would be my answer. (Maybe even this is ambiguous.)
Is there a way to ask it that leaves no room for ambiguity?

 

[Steve4Physics]

Is there a way to ask it that leaves no room for ambiguity?

I'll give it a go...

What is the maximum number of times ($N$ ) a circle* can intersect the graph of $y=\sin x$?
a)$N \le 2$ b) $N\le 4$ c)$N \le 6$ d) $N \le 8$ e)$N \gt 16$

*It’s implicit that all possible values for the centre and radius of the circle are permissble.

Edit - corrections

 

[FactChecker]

I would say ""out of all possible circles, what is the maximum number of times any circle can intersect the sine function."
IMO, that makes it explicit that all circles are to be considered.

 

[vcsharp2003]

Is anyone else appalled by the syntax of this question? Is (e) a correct answer? I cannot decide.
@vcsharp2003 What is the book and who publishes it ? Bring out the pitards!
I am reminded of Prof. Feynman's tale: https://rangevoting.org/FeynTexts.html

Its a Math book titled Mathematics 11 that is used in Grade 11 in schools of Ontario, Canada. The publisher is McGraw-Hill Ryerson and it has multiple authors from various school boards of Ontario, Canada.

You can see a full legal copy of it at the Internet Archive using the following link: https://archive.org/details/mcgrawhillryerso0000unse_d3o2, but to view the whole book you need to first register on their webiste and then click on the Borrow button. The challenge problem that I posted can be found on p 420.

 

[FactChecker]

It's tricky to state that problem unambiguously. I wouldn't be too hard on them for any ambiguity.

 

[vcsharp2003]

It seems that none of the options given in the challenge problem under post#13 is true.
I counted the number of intersections in post#10 to be greater than 16.
I also counted the number of intersections as 2 in post#3 .

In that case, shouldn't this challenge problem be wrong?

 

[.Scott]

First off, sorry for becoming unavailable. My follow-on hint would have been to try b=r-1, r>10.

But regarding ambiguity - Here's how it was originally stated by the OP:

The question is asking for the number of different solutions to the following two equations.
$$y=\sin{x}$$
$$(x-a)^2 + (y-b)^2 = r^2$$

I suppose you could interpret that as "find the function of a, b, and r that will the return number of solutions".
And that is certainly a tractable problem.

But even without knowing what the original verbatim test question and 5 optional answers were, I presumed that we were looking for a count and a maximum at that.

There were two reasons for this:
1) If a function was expected, the norm would be to explicitly request a function - unless the context for a function is very compelling.
2) "Different" can be also be interpreted in several ways - but it suggests looking for a maximum.

But once the verbatim test question and the 5 options have been provided, I see no ambiguity. There is no direct English-to-Math translation, but "can" generally implies what is possible - and in this case a maximum would imply a range from 0 to that maximum (all that "can" be).
Also, all of the optional answers talk about counts, not functions. None of the options allow for "12", so we are not being asked about all possible solutions - we are being asked about a subset.
Does the option "at most 8 times" mean that "8" plays a special role in this problem? If not, then selected subsets could yield answers of only 0, 1, 2, 3, and 4. But in those cases, "at most 4 times" would equally apply. So, the combination of "at most 4 times" and "at most 8 times" very strongly suggest that they are referring to a situation where the "4" or "8" are tied to the solution. There is, however, only one "more than" option (ie, "more than 16 times"), so the "16" does not need to be special.

 

[FactChecker]

It seems that none of the options given in the challenge problem under post#13 is true.
I counted the number of intersections in post#10 to be greater than 16.
I also counted the number of intersections as 2 in post#3 .

In that case, shouldn't this challenge problem be wrong?

It is only wrong if you assume the most trivial interpretation of the question. I would not do that.

Suppose you interpret the question to be: "Given a circle, what is the maximum number of intersections?".
Then you can be given a circle like $x^2+(y+1000)^2=1$, with no intersections. I think that is too trivial to be the correct interpretation.

On the other hand, suppose you interpret the question to be: "Given all possible circles, what is the largest number of intersections that any of them have?"
Then e: More than 16 is the correct answer.

 

[hutchphd]

Then e: More than 16 is the correct answer.

i still decry thae question.
More than 16 is a (not the!) correct answer. I think it unwise to include an example whose complications are almost entirely semantic (although this discussion, if engendered, would be educational to potential writers of textbooks. ) The math content rapidly becomes overwhelmed however.

 

[FactChecker]

i still decry thae question.
More than 16 is a (not the!) correct answer. I think it unwise to include an example whose complications are almost entirely semantic (although this discussion, if engendered, would be educational to potential writers of textbooks. ) The math content rapidly becomes overwhelmed however.

I agree. But I don't think the authors realized the ambiguity, so I am not inclined to be very critical of them.

 

[hutchphd]

They are writing a textbook. It is their job to worry about clarity. A
nd where were the editors.....? I am very critical. More so because this example is purely of their own choosing

 

[FactChecker]

They are writing a textbook. It is their job to worry about clarity. A
nd where were the editors.....? I am very critical. More so because this example is purely of their own choosing

Proofreading a book is a tedious, endless, thankless, job. No matter how long and hard you try, some errors remain. IMO, there must be something demonic going on. ;-)