Percent Difference Between Theta and Sin (Theta) help

[Nomad5133]

Homework Statement

This is what my professor is asking me to do:
Show a table for when sin(theta) and theta have a percent difference of 1%, 5%, and 10%. (i.e. this table will have three columns)" You are to show this table (for all three quantities) such that the behavior leading slightly up to and slightly after the 1%, 5% and 10% difference marks can be seen. You must have at least 5 points before and after each of these percent difference values, and your resolution should be small enough so that you can see the transition of the percent difference to these values (and then after them)
How would i go about doing this?! It makes no sense to me

Homework Equations

percent difference = ((Num1-Num2)/Num2) * 100

The Attempt at a Solution

I attempted to plug in the theta in which we measured in our class (5 degrees) and sin (5 degrees) and find the percent difference but then i realized he isn't asking to use this 5 degrees. I think he is asking just theta as in any angle and its sin are 1 5 and 10% difference.

 

[Char. Limit]

[STRIKE]When does

$$\frac{sin(\theta) - \theta}{\theta} * 100$$

Equal 1? 5? 10? You can solve this analytically for some, I think, but it might be easier here to solve it numerically.[/STRIKE]

Disregard this.

 

[Nomad5133]

Yes your on the right track. By saying "Disregard this" I am guessing your still thinking? Thank you! Let me know if/when you get it, this is the only part of my homework I have left lol.

 

[Char. Limit]

Yes your on the right track. By saying "Disregard this" I am guessing your still thinking? Thank you! Let me know if/when you get it, this is the only part of my homework I have left lol.

My problem was that I found that

$$\frac{sin(\theta) - \theta)}{\theta} = \frac{sin(\theta)}{\theta} - 1$$

Is always negative for all theta. But maybe if you took the negative of that, and set THAT equal to 1, 5, and 10... then it might work.

 

[micromass]

I'm very bad with percentages, but don't you need to have that

$$\frac{100|\sin(\theta)-\theta|}{\theta}=1$$

It makes more sense to me with absolute values...

And I doubth very much that this can be solved analytically. You can easily calculate it with a computer though, by using some kind of fixed point property:

You need to find when $$\theta$$ satisfies

$$\frac{1}{99}\sin(\theta)=\theta$$

Define the function $$f(\theta)=\sin(\theta)/99$$. You then pick a suitable value $$\theta_0$$ and you calculate $$f(f(f(f(f(f(...f(\theta_0)...))))))$$ (the more times you apply f, the more precise it gets). The values that you obtain should be the right value of $$\theta$$...

 

[Nomad5133]

I'm very bad with percentages, but don't you need to have that

$$\frac{100|\sin(\theta)-\theta|}{\theta}=1$$

It makes more sense to me with absolute values...

And I doubth very much that this can be solved analytically. You can easily calculate it with a computer though, by using some kind of fixed point property:

You need to find when $$\theta$$ satisfies

$$\frac{1}{99}\sin(\theta)=\theta$$

Define the function $$f(\theta)=\sin(\theta)/99$$. You then pick a suitable value $$\theta_0$$ and you calculate $$f(f(f(f(f(f(...f(\theta_0)...))))))$$ (the more times you apply f, the more precise it gets). The values that you obtain should be the right value of $$\theta$$...

Yes sorry it should be in absolute value. Thanks, how would I pick a suitable value for $$\theta_0$$ ?

 

[micromass]

Picking the correct $$\theta_0$$ is quite tricky. The only way I know is by drawing the graphs of $$f(\theta)=\frac{100\sin(\theta)}{99}$$ and of $$g(\theta)=\theta$$. You then need to guess a bit where the to graphs intersect (this will be in quite a few places). Take this guess as $$\theta_0$$. By applying f over and over, the guess will converge to the fixpoint.

Another method you can apply is by considering the equation $$100\sin(\theta)-99\theta=0$$. You can solve this equation numerically by applying a method like Newton-Rhapson...

 

[Char. Limit]

Picking the correct $$\theta_0$$ is quite tricky. The only way I know is by drawing the graphs of $$f(\theta)=\frac{100\sin(\theta)}{99}$$ and of $$g(\theta)=\theta$$. You then need to guess a bit where the to graphs intersect (this will be in quite a few places). Take this guess as $$\theta_0$$. By applying f over and over, the guess will converge to the fixpoint.

Another method you can apply is by considering the equation $$100\sin(\theta)-99\theta=0$$. You can solve this equation numerically by applying a method like Newton-Rhapson...

Alternatively, you could use something like MATLAB or Mathematica to get an answer to any number of digits you require.

 

[Nomad5133]

Picking the correct $$\theta_0$$ is quite tricky. The only way I know is by drawing the graphs of $$f(\theta)=\frac{100\sin(\theta)}{99}$$ and of $$g(\theta)=\theta$$. You then need to guess a bit where the to graphs intersect (this will be in quite a few places). Take this guess as $$\theta_0$$. By applying f over and over, the guess will converge to the fixpoint.

Another method you can apply is by considering the equation $$100\sin(\theta)-99\theta=0$$. You can solve this equation numerically by applying a method like Newton-Rhapson...

Okay so I plugged in $$\theta$$ = 0.25 degrees and got it close to 1%. How would i do this for 5, and 10%?

 

[micromass]

Do thesame thing with 5 and 10, instead of 1...

 

[Nomad5133]

so for $$100\sin(\theta)-99\theta=0$$

when finding 5% would i change it to $$100\sin(\theta)-20\theta=0$$?

 

[micromass]

No, how did you get that?

You need to solve

$$\frac{100|\sin(\theta)-\theta|}{\theta}=5$$

What does that give you?

 

[Nomad5133]

No, how did you get that?

You need to solve

$$\frac{100|\sin(\theta)-\theta|}{\theta}=5$$

What does that give you?

I simplified $$\frac{100|\sin(\theta)-\theta|}{\theta}=5$$ to be $$20\sin(\theta)-21\theta=0$$ and I plugged this into mathematica (WolframAlpha) and it doesn't give me a number besides the integer 0

 

[micromass]

You simplified it wrong... Specifically, the number 21 is not correct...

 

[Nomad5133]

You simplified it wrong... Specifically, the number 21 is not correct...

How? I assumed theta is positive. i set it up as 100 * (sin (theta) - theta) = 5 *theta

then sin (theta) - theta = theta/20

then 20 sin (theta) - 20 (theta) = theta

then 20 sin (theta) - 21 (theta) = 0

 

[micromass]

i set it up as 100 * (sin (theta) - theta) = 5 *theta

This is were your mistakes lies. The absolute value signs are quite important. The correct set-up is

$$100|\sin(\theta)-\theta|=5\theta$$

Eliminating the absolute value signs will either give you

$$100(\sin(\theta)-\theta)=5\theta$$

or

$$100(\theta-\sin(\theta))=5\theta$$

But this depends on whether $$\theta-\sin(\theta)$$ is positive...

 

[Nomad5133]

This is were your mistakes lies. The absolute value signs are quite important. The correct set-up is

$$100|\sin(\theta)-\theta|=5\theta$$

Eliminating the absolute value signs will either give you

$$100(\sin(\theta)-\theta)=5\theta$$

or

$$100(\theta-\sin(\theta))=5\theta$$

But this depends on whether $$\theta-\sin(\theta)$$ is positive...

Okay thank you I found an estimate of the three values for when its closest to 1%, 5%, and 10% now i am still confused on this part:
"You are to show this table (for all three quantities) such that the behavior leading slightly up to and slightly after the 1%, 5% and 10% difference marks can be seen. You must have at least 5 points before and after each of these percent difference values, and your resolution should be small enough so that you can see the transition of the percent difference to these values (and then after them)"

Do you understand what he wants me to do?What about in terms of the actual table he wants me to create?

 

[micromass]

Okay thank you I found an estimate of the three values for when its closest to 1%, 5%, and 10% now i am still confused on this part:
"You are to show this table (for all three quantities) such that the behavior leading slightly up to and slightly after the 1%, 5% and 10% difference marks can be seen. You must have at least 5 points before and after each of these percent difference values, and your resolution should be small enough so that you can see the transition of the percent difference to these values (and then after them)"

Do you understand what he wants me to do?What about in terms of the actual table he wants me to create?

Well, I can make a guess to what he wants. I think he wants you to find 5 terms such that the percentage difference between theta and sin(theta) is within 1%, and he probably wants a table with these 5 elements. Furthermore, I think the elements should be chosen in that way such that the trend becomes clear.

But I suggest asking your professor, I can only guess what he wants, and since I didn't get this assignment, it could be that I'm very wrong about this...

 

[D H]

Okay so I plugged in $$\theta$$ = 0.25 degrees and got it close to 1%. How would i do this for 5, and 10%?

You almost certainly are supposed to be working in radians here, not degrees. Your result of 0.25 is more-or-less correct if theta is in radians.

A better value than 0.25 is 0.245317809 radians, not degrees.

 

[Nomad5133]

Well, I can make a guess to what he wants. I think he wants you to find 5 terms such that the percentage difference between theta and sin(theta) is within 1%, and he probably wants a table with these 5 elements. Furthermore, I think the elements should be chosen in that way such that the trend becomes clear.

But I suggest asking your professor, I can only guess what he wants, and since I didn't get this assignment, it could be that I'm very wrong about this...

Okay I asked a couple of classmates and they say he wants to see the transition of when it gets close to 1%
Example:
when theta = 0.17 then 0.19, 0.20, 0.21, 0.23 its getting close to the percent difference being 1% (when theta = 0.25 it is about 1%)

How would i go about putting this in a table with only 3 columns?! (he suggests the table will have 3 columns)

 

[micromass]

How would i go about putting this in a table with only 3 columns?! (he suggests the table will have 3 columns)

I'm sorry, but I cannot help you with this. I didn't get the assignment, so I certainly wouldn't know what your professor means or what you're supposed to do. Maybe somebody else can help, but you'd be better off asking your professor or classmates...

 

[Nomad5133]

I'm sorry, but I cannot help you with this. I didn't get the assignment, so I certainly wouldn't know what your professor means or what you're supposed to do. Maybe somebody else can help, but you'd be better off asking your professor or classmates...

Okay thanks for all your help! Much appreciated

 

[SammyS]

Homework Statement

This is what my professor is asking me to do:
Show a table for when sin(theta) and theta have a percent difference of 1%, 5%, and 10%. (i.e. this table will have three columns)" You are to show this table (for all three quantities) such that the behavior leading slightly up to and slightly after the 1%, 5% and 10% difference marks can be seen. You must have at least 5 points before and after each of these percent difference values, and your resolution should be small enough so that you can see the transition of the percent difference to these values (and then after them)
How would i go about doing this?! It makes no sense to me

Homework Equations

percent difference = ((Num1-Num2)/Num2) * 100

The Attempt at a Solution

I attempted to plug in the theta in which we measured in our class (5 degrees) and sin (5 degrees) and find the percent difference but then i realized he isn't asking to use this 5 degrees. I think he is asking just theta as in any angle and its sin are 1 5 and 10% difference.

For small values of θ, i.e. θ << 1, sin(θ) ≈ θ, when θ is in radians !

I'm pretty sure that's what your prof. had in mind.

Find  $$\frac{\theta-\sin(\theta)}{\theta}\times 100\%\percent$$

You could use Wolfram Alpha to find θ for 1%, 5%, and 10%. Here's http://www.wolframalpha.com/input/?i=(x-sin(x))/x=.05+" Look down the page for the numerical root.