Finding the maximum and minimum points of a cosine function

Thanks!In summary, the conversation discussed solving a mathematical equation to find the maximums and minimums in terms of x. The equation was simplified using trigonometric identities and a derivative was taken to find the extrema. The resulting equation was solved using trigonometric functions and the solution was compared to the answer given in the text. The conversation also touched on the period of a function and the transformation of functions.
  • #1
VinnyCee
489
0

Homework Statement



This is part of a larger engineering problem, I have reduced it to this mathematical equation, which should be simple, right?

I need to find when the following EQ has Maximum's (in terms of x):

[tex]\left|4\,cos\left(\frac{2\pi}{5}\,-\,30\,X\right)\,-\,4\,cos\left(\frac{2\pi}{5}\,+\,30\,X\right)\right|[/tex]

Homework Equations

The Attempt at a Solution



I guess take a derivative to find when the maximum and minimums are...

[tex]120\,sin\left(\frac{2\pi}{5}\,-\,30\,X\right)\,+\,120\,sin\left(\frac{2\pi}{5}\,+\,30\,X\right)[/tex]

set that equal to zero and solve for X, so I get the following equation...

[tex]-sin\left(\frac{2\pi}{5}\,-\,30\,X\right)\,=\,sin\left(\frac{2\pi}{5}\,+\,30\,X\right)[/tex]

I can't solve that!

I put it into maple, this is what it gave...

[tex]-\frac{\pi}{300}\,-\frac{1}{30}\,arctan\left[4\,cos\left(\frac{3\pi}{10}\right)\,sin\left(\frac{3\pi}{10}\right)\,+\,2\,cos\left(\frac{3\pi}{10}\right)\right]\,\,=\,\,-0.0524[/tex]The answer is given in the text, but I can't seem to arrive at the same conclusion!

Supposedly, the answer (for the maxima) is...

[tex]X\,=\,\frac{\pi}{60}\,+\,\frac{2\,n\,\pi}{30}[/tex]

and for the minima...

[tex]X\,=\,\frac{n\pi}{30}[/tex]

but when I solve the original EQ (before taking the derivative) by graphing to find the max's - it's 1.1 - 3.3 - 5.5 - etc. which is a factor of 5 off from the book answer just above! not sure how to proceed from here!
 
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  • #2
VinnyCee said:
I need to find when the following EQ has Maximum's (in terms of x):

[tex]\left|4\,cos\left(\frac{2\pi}{5}\,-\,30\,X\right)\,-\,4\,cos\left(\frac{2\pi}{5}\,+\,30\,X\right)\right|[/tex]

Question: are there supposed to be absolute value signs on this function?

I'd make the suggestion of factoring the "4" out and then applying the sum-to-product trig identity

cos A - cos B = -2 · sin[ (A+B)/2 ] · sin[ (A-B) / 2 ] .

I believe this will leave you with a product where one of the terms now becomes a constant and you just have one trig function to deal with. That may make it easier to look for the extrema...
 
  • #3
Yes, the absolute value is part of the function. I'm not too worried about that though.

I converted the function to the simpler one:

[tex]\left|-8\,sin\left(\frac{2\pi}{5}\right)\,sin\left(-30\,X\right)\right|[/tex]

But how do I solve for extrema now?
 
  • #4
Since [tex]\,sin\left(\frac{2\pi}{5}\right)[/tex] is just a number, you can turn this into

[tex]8\,sin\left(\frac{2\pi}{5}\right)\left|\,sin\left(-30\,X\right)\right|[/tex].

The function inside the bars is equal to

[tex]-\,sin\left(30\,X\right)[/tex] ,

since sine is an odd function. The absolute value will eliminate the minus sign, leaving you with

[tex]8\,sin\left(\frac{2\pi}{5}\right)\left|\,sin\left(30\,X\right)\right|[/tex] ,

which is the absolute value of a sine function with period 2(pi)/30 . All the "valleys" going down to -1 now become "peaks", doubling the frequency of the maxima.

Now, where does sin(30x) have maxima and minima?
 
  • #5
I've no idea how to get an expression for the maxima and minima of an EQ. I forgot!

We know the minima will be zero...

[tex]0\,=\,sin(30\,X)[/tex]

take inverse sine? So...

[tex]X\,=\,\frac{sin^{-1}(0)}{30}[/tex]
 
  • #6
VinnyCee said:
I've no idea how to get an expression for the maxima and minima of an EQ. I forgot!

We know the minima will be zero...

[tex]0\,=\,sin(30\,X)[/tex]

take inverse sine? So...

[tex]X\,=\,\frac{sin^{-1}(0)}{30}[/tex]

Ah, but we won't be using the function arcsin(x); we just want to know where sin(30x) is either zero or has its maximum value of 1. There will be an infinite number of solutions.

There is a minimum at x = 0, as you say. The next one (to the right) will be half a period away. Since the period is 2(pi)/30, that would put it at x = (1/2)·2(pi)/30 = (pi)/30. The rest will be successive half-periods further along.

Where will the first maximum to the right of x = 0 be?
 
  • #7
The first maxima would be half way between the first and second minima, right?

So the first maxima occurs at [itex]\frac{\pi}{60}[/itex]? And then at every integer multiple of the period afterwards?

[tex]X\,=\,\frac{\pi}{60}\,+\,\frac{2\,n\,\pi}{30}[/tex]

It's right! So how did you get that the period of the sin(30x) was [itex]\frac{2\pi}{30}[/itex]? I suppose it's a fundamental EQ where the period is just 2[itex]\pi[/itex] divided by the coefficient of the x term within the sin expression, right?Thanks!
 
  • #8
VinnyCee said:
The first maxima would be half way between the first and second minima, right?

So the first maxima occurs at [itex]\frac{\pi}{60}[/itex]? And then at every integer multiple of the period afterwards?

[tex]X\,=\,\frac{\pi}{60}\,+\,\frac{2\,n\,\pi}{30}[/tex]

This is the one part of the given solution I was puzzled by. If you taking the absolute value, the "valleys" half a period away should become "peaks", so there would be maxima every half-period along, which would be [tex]\frac{\,n\,\pi}{30}[/tex]. That's partly why I was asking if the absolute value signs were supposed to be there; I'm wondering if the person who wrote the answer forgot about them.

So how did you get that the period of the sin(30x) was [itex]\frac{2\pi}{30}[/itex]? I suppose it's a fundamental EQ where the period is just 2[itex]\pi[/itex] divided by the coefficient of the x term within the sin expression, right?

The function sin(kx) can be seen as a transformation of sin(x) which involves "horizontal compression" about the y-axis by a factor of k (for k > 1). Since sin(x) has a period of 2(pi) , the function sin(kx) will have period 2(pi)/k .
 
  • #9
[itex]\frac{n\pi}{30}[/itex] is the text answer for the minima.

The function value at the minima and maxima is 0 and ~7.6085, respectively.
 
  • #10
VinnyCee said:
[itex]\frac{n\pi}{30}[/itex] is the text answer for the minima.

Yes, there will be two minima per period of 2(pi)/30, which correspond to the zeroes of sin(30x). For |sin(30x)|, there should also be two maxima per period, corresponding to the location of +/-1 for sin(30x), which is why I think something was overlooked in the printed answer.

The function value at the minima and maxima is 0 and ~7.6085, respectively.

That checks: [tex]8\,sin\left(\frac{2\pi}{5}\right)[/tex] is indeed ~7.6085.
 

What is a cosine function?

A cosine function is a mathematical function that relates the cosine of an angle to the length of the adjacent side and the length of the hypotenuse in a right triangle.

How do you find the maximum and minimum points of a cosine function?

To find the maximum and minimum points of a cosine function, you can use calculus to take the derivative of the function and set it equal to zero. The resulting values will be the x-coordinates of the maximum and minimum points.

What is the significance of the maximum and minimum points in a cosine function?

The maximum and minimum points in a cosine function represent the highest and lowest values that the function can reach. They are important in understanding the behavior and characteristics of the function.

Can a cosine function have more than one maximum or minimum point?

Yes, a cosine function can have multiple maximum and minimum points depending on its amplitude, period, and other parameters. These points will occur at regular intervals and can be found by solving the derivative equation.

How can the maximum and minimum points of a cosine function be used in real-life applications?

The maximum and minimum points of a cosine function can be used to analyze and predict the behavior of natural phenomena such as sound waves, light waves, and oscillating systems. They are also commonly used in fields such as engineering, physics, and astronomy to model and understand various physical processes.

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