Find Central Angle: Newton's Method

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In summary, the conversation discusses finding the central angle and arc length using Newton's Method and the Law of Cosines. The solution involves forming an equation using the Law of Cosines and using Newton's Method to approximate the root. The final answer is approximately 2.2622 radians or 130 degrees. The use of the ans feature on a calculator is also mentioned.
  • #1
karush
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In the figure, the length of the chord $AB$ is $4 \text { cm}$ and the length of the arc is $5\text{ cm}$
https://www.physicsforums.com/attachments/1713

(a) Find the central angle $\theta$, in radians, correct to four decimal places.

(b) Give the answer to the nearest degreethis problem is intended to be solved by Newton's Method, so I am have ?? as to how to set it up. I thot that using law of cosines would be part of it since

$$\cos{\theta} = \frac{4^2}{a^2+b^2-2ab}$$

or since $a=b=r$

$$\cos{\theta} = \frac{16}{2r^2-1}$$
and
$$\cos^{-1}{\left(\frac{16}{2r^2-1}\right)}=\theta$$

and also $$S=\theta\cdot r$$ for arc length

so this is where I have a flat tire without a spare...
 
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  • #2
The Law of Cosines gives you:

\(\displaystyle 4^2=r^2+r^2-2r^2\cos(\theta)\)

\(\displaystyle 8=r^2\left(1-\cos(\theta) \right)\)

Form the arc-length, we find:

\(\displaystyle 5=r\theta\implies r=\frac{5}{\theta}\)

And so, by substitution, we obtain:

\(\displaystyle 8=\left(\frac{5}{\theta} \right)^2\left(1-\cos(\theta) \right)\)

\(\displaystyle f(\theta)=8\theta^2+25\left(\cos(\theta)-1 \right)=0\)

Now use Newton's method to find the root for which \(\displaystyle 0<\theta\).
 
  • #3
View attachment 1724

(a) $2.2622$ rad
(b) $130^o$

I hope this got it...
 
  • #4
I would have looked at a graph of the function:

View attachment 1726

From this we see the root we seek is about \(\displaystyle x\approx2.25\).

Newton's method gives us the recursive algorithm:

\(\displaystyle x_{n+1}=x_n-\frac{8x_n^2+25\left(\cos\left(x_n-1 \right) \right)}{16x_n-25\sin\left(x_n \right)}\)

Now, on my TI-89, I enter the following:

InputOutput
2.25 [ENTER]2.25
ans(1)-(8ans(1)^2+25(cos(ans(1))-1))/(16ans(1)-25sin(ans(1))) [ENTER]2.26234822745
[ENTER]2.2622051906
[ENTER]2.2622051713
[ENTER]2.2622051713

And in degrees, this is about \(\displaystyle 129.614808708^{\circ}\).
 

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  • #5
well that a good thing to know, tried on my TI-nspire cx cas and got the same thing.
good use of the ans feature.:cool:
 
  • #6
karush said:
well that a good thing to know, tried on my TI-nspire cx cas and got the same thing.
good use of the ans feature.:cool:

Yes, that function allows for easy evaluation of the terms in a recursive algorithm. For example, you can get the Fibonacci numbers with:

0 [ENTER]
1 [ENTER]
ans(1)+ans(2) [ENTER]
Now press [ENTER] for each new number in the sequence. :D
 

FAQ: Find Central Angle: Newton's Method

What is Newton's Method and how is it used to find central angles?

Newton's Method is a mathematical algorithm used to approximate the roots of a given function. In the context of finding central angles, it can be used to approximate the angle between two intersecting lines or the angle between a tangent line and a curve. It relies on an iterative process that involves finding the tangent line to a point on the curve and using that tangent line to generate a new point closer to the root of the function.

What are the advantages of using Newton's Method to find central angles?

One of the main advantages of using Newton's Method is its efficiency and accuracy. It typically requires fewer iterations than other methods, such as the bisection method, to find a solution. Additionally, it can provide a more precise approximation of the root, making it a useful tool for solving complex mathematical problems.

What are the limitations of using Newton's Method to find central angles?

One limitation of Newton's Method is that it may fail to converge or may converge to a wrong solution if the initial guess is not close enough to the actual root. This is known as the "basin of attraction" problem. Additionally, it may not be suitable for functions with multiple roots or singularities, as it may converge to a local minimum or maximum instead of the desired root.

Can Newton's Method be applied to any type of function to find central angles?

No, Newton's Method is most effective when applied to continuous and differentiable functions. It may not work for discontinuous or non-differentiable functions. Additionally, the function must have at least one root in the given interval for Newton's Method to converge to a solution.

How does the choice of initial guess affect the accuracy of Newton's Method for finding central angles?

The choice of initial guess is crucial in determining the accuracy and convergence of Newton's Method. A good initial guess that is close to the actual root will result in faster convergence and a more accurate approximation. On the other hand, a poor initial guess may result in the method failing to converge or converging to a wrong solution. Therefore, it is important to choose an initial guess that is informed by the nature of the function and the given problem.

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