Solutions to Cos 2x Homework: 32 Answers

In summary: We are all 100% sure that he is right and the teacher is wrong, but how can the teacher - who is 100% wrong - be told without creating a problem? Maybe the OP should avoid any confrontation and simply proceed in his/her own way - we are not talking about any fundamental misunderstanding here. [QUOTE = "Simon Bridge, post: 5481001, member: 133405"]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
  • #1
takando12
123
5

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?
 
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  • #2
takando12 said:

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 ?
 
  • #3
takando12 said:

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.
 
  • #4
Once you plot in Wolfram Alfa, you can always print from your browser, like the attached image
 

Attachments

  • cos2xplot.pdf
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  • #5
SteamKing said:
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
 
  • #6
Thank you all for the help, I shall approach the teacher with some confidence now :)
 
  • #7
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.
 
  • #8
takando12 said:
...

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.​
 
  • #9
Merlin3189 said:
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.
 
  • #10
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.
 
  • #11
Merlin3189 said:
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.
 

FAQ: Solutions to Cos 2x Homework: 32 Answers

What is the concept of cos 2x?

The concept of cos 2x refers to the cosine function where the argument is multiplied by 2. In other words, it is the cosine of twice the angle.

How do you find solutions to cos 2x equations?

To find solutions to cos 2x equations, you can use trigonometric identities such as double angle formula or half angle formula. You can also graph the equation or use a calculator to find the values of x that satisfy the equation.

Can you provide an example of solving a cos 2x equation?

Yes, for example, if we have the equation cos 2x = 1, we can use the double angle formula cos 2x = 2cos^2x - 1 to rewrite it as 2cos^2x - 1 = 1. Solving for cos^2x, we get cos^2x = 1, which means cos x = ±1. Therefore, the solutions are x = 0 + 2πn and x = π + 2πn, where n is an integer.

What are the important properties of cos 2x?

Some important properties of cos 2x include: it is an even function, meaning cos 2x = cos (-2x); it has a period of π, meaning cos 2x repeats every π units; and its range is [-1,1], meaning the values of cos 2x can never be greater than 1 or less than -1.

How is cos 2x used in real-life applications?

Cos 2x has various applications in fields such as engineering, physics, and astronomy. For example, it is used in calculating vibrations and oscillations, determining the position of celestial objects, and analyzing alternating current circuits. It is also used in computer graphics and animation to create rotating and oscillating movements.

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