- #1
Grand
- 74
- 0
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.
Equation of circle with arc length
- Thread starter Grand
- Start date
-
- Tags
- Arc Arc length Circle Length
Click For Summary
SUMMARY
The discussion centers on solving the equation of a circle given by x² + (y + a)² = R², with x-intercepts at ±√3 and an arc length above the x-axis of 4π/3. Participants suggest that while the equation can be approached using arcsin and numerical methods, a more straightforward solution can be achieved through guess-and-check combined with diagrammatic representation. The consensus is that a rigorous method may not be necessary, as intuitive methods can yield results quickly.
PREREQUISITES- Understanding of circle equations in Cartesian coordinates
- Familiarity with arc length calculations in geometry
- Basic knowledge of trigonometric functions, specifically arcsin
- Experience with numerical approximation methods
- Explore the derivation of arc length formulas in circular geometry
- Learn about the properties of sine functions and their inverses
- Investigate numerical methods for solving equations, such as Newton's method
- Study the application of guess-and-check methods in mathematical problem-solving
Students studying geometry, mathematics educators, and anyone interested in problem-solving techniques for equations involving circles and trigonometric functions.
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.
- #2
tiny-tim
Science Advisor
Homework Helper
- 25,837
- 258
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!
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!
- #3
Grand
- 74
- 0
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.
\arcsin{\frac{\sqrt{3}}{R}}=\frac{4\pi/3}{2R}
I can't solve it, and haven't found anything else useful.
- #4
tiny-tim
Science Advisor
Homework Helper
- 25,837
- 258
Hello Grand!
Yes, that looks good …
but it's obviously unsolvable without a computer, sooo
…
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? 
- #5
Grand
- 74
- 0
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?
- #6
tiny-tim
Science Advisor
Homework Helper
- 25,837
- 258
hmm … computer, Newtonian approximation, bribing the TA …
… computer, Newtonian approximation, bribing the TA … - #7
Grand
- 74
- 0
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?
- #8
tiny-tim
Science Advisor
Homework Helper
- 25,837
- 258
uhhh? 
how many angles do you know with "√3" in the sine ?
- #9
Grand
- 74
- 0
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.
- #10
eumyang
Homework Helper
- 1,347
- 11
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.