Equation of circle with arc length

[Grand]

Homework Statement

Equation of a circle is:
$$x^2+(y+a)^2=R^2$$

x intercepts are $$+/-\sqrt{3}$$ and arclength above x-axis is $$\frac{4\pi}{3}$$

Find a and R.

 

[tiny-tim]

Welcome to PF!

Hi Grand! Welcome to PF!

(have a square-root: √ and a pi: π and try using the X² icon just above the Reply box )

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[Grand]

Hello, I've been tackling the problem for a while and the equation that I managed to come up with is:
$$\arcsin{\frac{\sqrt{3}}{R}}=\frac{4\pi/3}{2R}$$

I can't solve it, and haven't found anything else useful.

 

[tiny-tim]

Hello Grand!

Yes, that looks good …

but it's obviously unsolvable without a computer, sooo …

hadn't you better assume the answer is really obvious, and just make a guess and see if it's right? ​

 

[Grand]

Well, is it? Even though it is stated in the book, I want to find a way to actually obtain it. Is there a way?

 

[tiny-tim]

hmm … computer, Newtonian approximation, bribing the TA …

 

[Grand]

Taylor was my guess too, but here we aim at an exact answer, so we should not use it. Even though, how would you guess the answer?

 

[tiny-tim]

uhhh?

how many angles do you know with "√3" in the sine ? ​

 

[Grand]

Yeah, alright. But come on, there must be some rigorous way doing it - I've been solving it for couple of hrs now, even managed to prove Pythagoras with it.

 

[eumyang]

What's wrong with guess-and-check? It's a valid method. I don't mean to put you down, but combining this method with drawing a diagram, I was able to get the answer within minutes.