Need help w/ expanding cosine fcn: (1/2)theta_1^2 + cos(theta_1) - cos(theta) =

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dhkstyle

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Hello, I was looking at a website that models the time it takes for a pencil to fall over when it's balanced on its tip.
This is the URL of the page: http://www.mathpages.com/home/kmath259/kmath259.htm
Towards the bottom of the page, I encountered a part that I have difficulty understanding. It says:
(1/2)theta12 + cos(theta1) - cos(theta) = (1-cos(theta)) + (1/24)theta14 + (1/720)theta16 + ...
where theta1 is the initial angle, and theta is the angular position
How did the person expand the expression? Is this expansion a manipulation of the Taylor series? If so, can someone please help me through the steps in this expansion?
Also, right below it says "Since we are taking a small value of theta1, the higher order terms can be neglected, and the argument of the square root is simply 1 - cos(theta) = 2sin(theta/2)2. Making this substitution, the integral can be evaluated in closed form as... (I couldn't type out the integral)
I don't understand why 1 - cos(theta) = 2sin(theta/2)2 and how the integral below that sentence was simplified.
Sorry for asking so many questions, but I would really appreciate if someone can help me understand this part. Thank you in advance
 
dhkstyle said:
Hello, I was looking at a website that models the time it takes for a pencil to fall over when it's balanced on its tip.
This is the URL of the page: http://www.mathpages.com/home/kmath259/kmath259.htm
Towards the bottom of the page, I encountered a part that I have difficulty understanding. It says:
(1/2)theta12 + cos(theta1) - cos(theta) = (1-cos(theta)) + (1/24)theta14 + (1/720)theta16 + ...
where theta1 is the initial angle, and theta is the angular position
How did the person expand the expression? Is this expansion a manipulation of the Taylor series? If so, can someone please help me through the steps in this expansion?
Also, right below it says "Since we are taking a small value of theta1, the higher order terms can be neglected, and the argument of the square root is simply 1 - cos(theta) = 2sin(theta/2)2. Making this substitution, the integral can be evaluated in closed form as... (I couldn't type out the integral)
I don't understand why 1 - cos(theta) = 2sin(theta/2)2 and how the integral below that sentence was simplified.
Sorry for asking so many questions, but I would really appreciate if someone can help me understand this part. Thank you in advance
Click to expand...
That is a basic trigonometric relation.

Do you know the angle addition rules like:

cos(A+B) = cos(A) * cos(B) - sin(A) * sin(B)
 
dhkstyle said:
Thank you! Could you also help me with the expansion part?
Click to expand...
You're referring to a lesson in physics. The lesson clearly specifies that what you've called an "equation" is actually an approximation, "very close to theta-sub-zero".

I would suspect that, yes, this relates to Taylor series and the like. Unfortunately, it is not reasonably feasible within this environment to attempt to teach a few weeks (at least!) of of calculus to you. Sorry. Please feel free to attempt online self-study. ;)
 
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