Fundraise on MathsGee
First time here? Checkout the FAQs!

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

1 like 0 dislike
The circle below has radius 1 , and the longer circular are joining \(A\) and \(B\) is twice as long as the chord \(A B\). Find the length of the chord \(A B\), correct to 18 decimal places.
in Mathematics by Platinum (130,878 points) | 417 views

1 Answer

0 like 0 dislike
Best answer

This is somewhat of a trick question. Sorry! It seemed like a good idea at the time.

Draw a perpendicular from \(O\) to \(A B\), meeting \(A B\) at \(M\). Let \(\theta=\) \(\angle A O M\). Standard trigonometry shows that the length of \(A B\) is \(2 \sin \theta\). The shorter arc joining \(A\) and \(B\) has length \(2 \theta\), so the longer arc has length \(2 \pi-2 \theta\). The longer arc is twice the chord, and therefore
2 \pi-2 \theta=4 \sin \theta .


We can use the Newton Method to solve this equation as it stands. Let \(f(\theta)=2 \sin \theta+\theta-\pi\). Then \(f^{\prime}(\theta)=2 \cos \theta+1\), and the Newton Method recurrence is
\theta_{n+1}=\theta_{n}-\frac{2 \sin \theta_{n}+\theta_{n}-\pi}{2 \cos \theta_{n}+1} .
An ordinary calculator will only handle this to 8 or 9 places. The scientific calculator that comes bundled with Microsoft Windows can handle about 30 .

But we can find the answer without doing any work by looking back at a calculation done in the notes. We are solving \(\pi-\theta=2 \sin \theta\). Note that by the symmetry of the sine function, we have \(\sin \theta=\sin (\pi-\theta)\). Let \(x=\pi-\theta\). Then our equation is equivalent to \(x=2 \sin x\).

It so happens that in the notes this equation is solved to high accuracy. The positive root of this equation, to 19 places, is given there as \(x=\) \(1.8954942670339809471\). But the length of the chord is \(2 \sin \theta\), that is, \(2 \sin x\), and that is equal to \(x\).

by Platinum (130,878 points)

Related questions

2 like 0 dislike
0 answers
asked Aug 8 in Mathematics by MathsGee Platinum (130,878 points) | 15 views
1 like 0 dislike
1 answer
0 like 0 dislike
2 answers
1 like 0 dislike
1 answer
asked May 15, 2021 in Mathematics by MathsGee Platinum (130,878 points) | 155 views

Join the MathsGee Learning Club where you get study and financial support for success from our community. CONNECT - LEARN - FUNDRAISE

On the MathsGee Learning Club, you can:

1. Ask questions

2. Answer questions

3. Vote on Questions and Answers

4. Start a Fundraiser

5. Tip your favourite community member(s)

6. Create Live Video Tutorials (Paid/Free)

7. Join Live Video Tutorials (Paid/Free)

8. Earn points for participating

Posting on the MathsGee Learning Club

1. Remember the human

2. Behave like you would in real life

3. Look for the original source of content

4. Search for duplicates before posting

5. Read the community's rules


1. Answers to questions will be posted immediately after moderation

2. Questions will be queued for posting immediately after moderation

3. Depending on how many posts we receive, you could be waiting up to 24 hours for your post to appear. But, please be patient as posts will appear after they pass our moderation.

MathsGee Android Q&A