Solutions to Cos 2x Homework: 32 Answers

[takando12]

Homework Statement

Cos 2x =√3/2 Find number of solutions .x∈[0,8π]

Homework Equations

The Attempt at a Solution

:[/B]
drew the graph of cos 2x.It has a periodicity of π and there are two solutions in each period. So 2*8 =16 solutions totally. But my teacher said that we must take [0,16π] as the interval because of 2x and said the answer is 32 solutions. I don't understand how this works. We are taking values of x on the x-axis of the graph , so why should we take 16π? How is it 32 solutions?

 

[SteamKing]

Homework Statement

Cos 2x =√3/2 Find number of solutions .x∈[0.8π]

I don't understand your notation here. It looks like 0.8*π ≈ 2.51

Do you mean find the angle x such that x ∈ [0 - 8π], or, in other words 0 ≤ x ≤ 8π ?

 

[Ray Vickson]

Homework Statement

Cos 2x =√3/2 Find number of solutions .x∈[0.8π]

Homework Equations

The Attempt at a Solution

:[/B]
drew the graph of cos 2x.It has a periodicity of π and there are two solutions in each period. So 2*8 =16 solutions totally. But my teacher said that we must take [0,16π] as the interval because of 2x and said the answer is 32 solutions. I don't understand how this works. We are taking values of x on the x-axis of the graph , so why should we take 16π? How is it 32 solutions?

Your post has a typo in it, but if you mean $x \in [0,8 \pi]$ then you are correct.

Your problem might be: how to put it diplomatically to your teacher? I suggest plotting a graph where you can actually see all the points and count them out. For example, you can do in in the (free) on-line package Wolfram Alpha, by entering the command
plot cos(2*x) and sqrt(3)/2 for x from 0 to 8*pi.
It produces a nice image, showing exactly 16 points of intersection, as you claim. So far, I cannot see how you could print the image, or save it for later printing or embedding in a document (Wolfram is not very helpful on that issue), but you can at least show the on-line results if asked for evidence. Maybe other helpers can tell you how to save or print the image.

 

[SteamKing]

Once you plot in Wolfram Alfa, you can always print from your browser, like the attached image

 

[takando12]

I don't understand your notation here. It looks like 0.8*π ≈ 2.51

Do you mean find the angle x such that x ∈ [0 - 8π], or, in other words 0 ≤ x ≤ 8π ?

yes , i edited it

 

[takando12]

Thank you all for the help, I shall approach the teacher with some confidence now :)

 

[Merlin3189]

Agree with all foregoing. And from the comment about 0 - 16π, just plain wrong.
And I know that by convention, √ always implies the positive root, but since the square root of 3 is +1.732 or -1.732, is it possible that the answer required was based on
cos 2x = +0.866 or -0.866 ? This would give 32 solutions.

Obviously WolframAlpha doesn't think so and I checked this with a maths teacher and was assured this could not be so and they would have to ask
cos 2x = +/- √3 /2 or maybe cos² 2x = 3/4
but it seems more plausible to me, than someone saying x∈[0,8π] means x∈[0,16π] when x is used in a function of 2x.

 

[SammyS]

...

drew the graph of cos 2x.It has a periodicity of π and there are two solutions in each period. So 2*8 =16 solutions totally. But my teacher said that we must take [0,16π] as the interval because of 2x and said the answer is 32 solutions. I don't understand how this works. We are taking values of x on the x-axis of the graph , so why should we take 16π? How is it 32 solutions?

What your teacher may have been thinking --

cos(x) has a period of 2π , with two solutions per period. That's 8 solutions in [0, 8π]. But since we have cos(2x) here, double that, so it's 16 solutions.​

 

[takando12]

but it seems more plausible to me, than someone saying x∈[0,8π] means x∈[0,16π] when x is used in a function of 2x.

i thought of that explanation too . But my teacher quite distinctly said and wrote on the board
cos x --- x[0,8π]
cos 2x --- x[0,16π] and then said since it's a periodicity of π 16*2=32. I shall ask him what he meant.

 

[Merlin3189]

I shall ask him what he meant.

That's for you to judge: depends what he's like. And whether you really want to know what he thinks! If he just says, that's the way it is and we're wrong, you'll just have to make up your own mind.

In my time teaching I can remember making a few gaffes, that no one ever queried. When I later realized my errors (and of course, I don't know about any I didn't later realize) I always felt deeply embarrassed, so I'm not sure how I would have reacted if someone had pointed them out. I hope I would have been suitably humble, grateful and have acknowledged that we can all make mistakes.

 

[Ray Vickson]

That's for you to judge: depends what he's like. And whether you really want to know what he thinks! If he just says, that's the way it is and we're wrong, you'll just have to make up your own mind.

In my time teaching I can remember making a few gaffes, that no one ever queried. When I later realized my errors (and of course, I don't know about any I didn't later realize) I always felt deeply embarrassed, so I'm not sure how I would have reacted if someone had pointed them out. I hope I would have been suitably humble, grateful and have acknowledged that we can all make mistakes.

All of what you said is the background reason that I suggested a graphical explanation in Post #3. That would be a positive approach ("here are the results, all plotted out") rather than the negative approach ("you have made an error"). Even so, the OP will be in an uncomfortable position.