how do i swap it round so i get the degrees for angle A?

[neilo]

It might seem like a trigonometry question, but its the manipulating of the formula i'm struggling with



Im not a student so its not homework by the way. I want to know how to express the above formula so i can get the angle (A) from the chord (a) 0.78 and both sides are 2 units in length in this example.

below is what i found on wolfram website, but i cant figure out how to turn it around. I might be just putting it in the calculator wrongly, so maybe adice on what buttons to press would be helpful too. Thanks

A circle chord is a line segment whose endpoints lie on the circle.
Chord length equals twice the radius times the sine of half the angle covered by the chord.

Formula​

 

[logistic_guy]

It might seem like a trigonometry question, but its the manipulating of the formula i'm struggling with

View attachment 39189

Im not a student so its not homework by the way. I want to know how to express the above formula so i can get the angle (A) from the chord (a) 0.78 and both sides are 2 units in length in this example.

below is what i found on wolfram website, but i cant figure out how to turn it around. I might be just putting it in the calculator wrongly, so maybe adice on what buttons to press would be helpful too. Thanks

A circle chord is a line segment whose endpoints lie on the circle.
Chord length equals twice the radius times the sine of half the angle covered by the chord.

Formula​

\(\displaystyle a = 0.78036129\)
\(\displaystyle r = 2\)
\(\displaystyle A = 2\sin^{-1}\left(\frac{a}{2r}\right) = 2\sin^{-1}\left(\frac{0.78036129}{2\times 2}\right) = 0.39269908 \ \text{rad} = 22.5^{\circ}\)

 

[jonah2.0]

Beer assisted reckoning follows.

It might seem like a trigonometry question, but its the manipulating of the formula i'm struggling with

View attachment 39189

Im not a student so its not homework by the way. I want to know how to express the above formula so i can get the angle (A) from the chord (a) 0.78 and both sides are 2 units in length in this example.

below is what i found on wolfram website, but i cant figure out how to turn it around. I might be just putting it in the calculator wrongly, so maybe adice on what buttons to press would be helpful too. Thanks

A circle chord is a line segment whose endpoints lie on the circle.
Chord length equals twice the radius times the sine of half the angle covered by the chord.

Formula​

\(\displaystyle a = 0.78036129\)
\(\displaystyle r = 2\)
\(\displaystyle A = 2\sin^{-1}\left(\frac{a}{2r}\right) = 2\sin^{-1}\left(\frac{0.78036129}{2\times 2}\right) = 0.39269908 \ \text{rad} = 22.5^{\circ}\)

Not familiar with that formula but the law of cosines works just as well.
See row #9

neilo how do i swap it round so i get the degrees for angle A? without cord

www.desmos.com

 

[Dr.Peterson]

i cant figure out how to turn it around. I might be just putting it in the calculator wrongly, so maybe advice on what buttons to press would be helpful too.

What did you put into the calculator? And what did you get that you knew was wrong?

What kind of calculator are you using? That will affect what buttons to press.

Are you familiar with the idea of inverse trig functions? Or inverse functions in general?

And was the calculator set to degrees or radians? Which do you want to get?

Not familiar with that formula but the law of cosines works just as well.

The formula is for an isosceles triangle, which is what you get for a chord a if the center is at A. It's based on dropping an altitude from A, forming two congruent right triangles.

 

[logistic_guy]

Beer assisted reckoning follows.



Not familiar with that formula but the law of cosines works just as well.
See row #9

neilo how do i swap it round so i get the degrees for angle A? without cord

www.desmos.com

BeautifulSir jonah\(\displaystyle 2.0\)

 

[neilo]

thanks very much.
This works great. I've been looking for this solution for 20 + years. I kept giving up on it, but it would keep bothering me.

 

[jonah2.0]

Beer drenched reaction follows.

thanks very much.
This works great. I've been looking for this solution for 20 + years. I kept giving up on it, but it would keep bothering me.

20 + years!
It's beautiful, really.
It's like a pearl.

Return to mathematical circles : a fifth collection of mathematical stories and anecdotes : Eves, Howard Whitley, 1911- : Free Download, Borrow, and Streaming : Internet Archive

xxi, 181 p. : 24 cm

archive.org

 

[neilo]

thanks both of you, I got it sussed now from logistic guys first comment. Thank you very much for compiling the graph solution for me . I will take a good look at it next weekend. What you showed me already is a massive help.
I can do it with my casio calculator fine, I did have trouble with other radii than 2 at first as I divided by r x r but I ammended my notes to be r+r.
Fantastic

Yes, the inverse sine was hindering me and not knowing how to set the problem out really.

 

[logistic_guy]

thanks very much.
This works great. I've been looking for this solution for 20 + years. I kept giving up on it, but it would keep bothering me.

You're welcome!



Since professor Dave mentioned it, I would comment on it.
Be careful when calculating angles with your calculator! Many students forget the radian set up on when they want the degree angle.

 

[jonah2.0]

Beer induced reflection follows.

... Not familiar with that formula but ...

Not exactly sure what happened there but I guess getting frustrated with mobile Excel while heavily intoxicated and sleep deprived can make one forget stuff. Good thing Dr. Peterson gave me a gentle reminder; I was looking at neilo's diagram and quickly dismissed the indicated formula in favor of the cosine law. The moral of the story is that One can drink and DERIVE but should never EVER drink and Drive. One can walk away from the former but could have life altering consequences if one foolishly insisted with the latter. And yet people still do drink and Drive. A little Desmos indulgence,

neilo how do i swap it round so i get the degrees for angle A? Reoriented

www.desmos.com

\(\displaystyle 2\sin\left(\frac{A}{2}\right)=\frac{a}{2}\implies\sin\left(\frac{A}{2}\right)=\frac{a}{2}\cdot\frac{1}{2}\implies\frac{A}{2}=\sin^{-1}\left(\frac{a}{2}\cdot\frac{1}{2}\right)\implies A=2\sin^{-1}\left(\frac{a}{2}\cdot\frac{1}{2}\right)\)

as indicated by neilo's diagram and Mario's reckoning.