Circular trigonmetry:- How to prove these formulaes?

[Win_odd Dhamnekar]

Using the above figure, I want to prove the following formulaes.

1) The chord length is . In this formulaes, I don't know the proof of last formulae

2)The saggita is In this case, saggita is height of the segment. d is the height of triangular portion.

3)The angle is


Lastly, The area A of the circular segment \(A=R^2*\left(\arcsin{\frac{c}{2R}}-\frac{c}{2R}*\sqrt{1-(\frac{c}{2R})^2}\right)=R\left(\arccos\frac{d}{R}-d\sqrt{1-\frac{d^2}{R^2}}\right)\)

 

[Win_odd Dhamnekar]

Using the above figure, I want to prove the following formulaes.

1) The chord length is View attachment 18434 . In this formulaes, I don't know the proof of last formulae

2)The saggita is View attachment 18436 In this case, saggita is height of the segment. d is the height of triangular portion.

3)The angle is
View attachment 18439

Lastly, The area A of the circular segment \(A=R^2*\left(\arcsin{\frac{c}{2R}}-\frac{c}{2R}*\sqrt{1-(\frac{c}{2R})^2}\right)=R\left(\arccos\frac{d}{R}-d\sqrt{1-\frac{d^2}{R^2}}\right)\)

Hello,

If any member of this forum knows how to prove these formulas step by step may reply to this question. However, I am also trying to understand and prove these formulas with my simple logic.?

 

[Dr.Peterson]

Please take the formulas one at a time, and tell us (a) which of them you have already proved, and how (since parts may be reusable for other formulas); and (b) what you have tried for one of the others, so we can focus on that one. Trying to juggle 11 formulas at once would make this an impossible thread.

We need to see your thinking in order to help you use it!

 

[pka]

Using the above figure, I want to prove the following formulas.
1) The chord length is View attachment 18434 . In this formulaes, I don't know the proof of last formulae
2)The saggita is View attachment 18436 In this case, saggita is height of the segment. d is the height of triangular portion.
3)The angle is View attachment 18439
Lastly, The area A of the circular segment \(A=R^2*\left(\arcsin{\frac{c}{2R}}-\frac{c}{2R}*\sqrt{1-(\frac{c}{2R})^2}\right)=R\left(\arccos\frac{d}{R}-d\sqrt{1-\frac{d^2}{R^2}}\right)\)

I agree with all that Prof. Peterson wrote above. The webpage from which that image comes is a jumble if formulae.
I do have one suggestion. Add a line segment from the centre to the circle that bisects the angle \(\theta\).
That line is also the perpendicular bisector of the cord \(c\) has length \(R=h+d\).

 

[Deleted member 4993]

Using the above figure, I want to prove the following formulaes.

1) The chord length is View attachment 18434 . In this formulaes, I don't know the proof of last formulae

2)The saggita is View attachment 18436 In this case, saggita is height of the segment. d is the height of triangular portion.

3)The angle is
View attachment 18439

Lastly, The area A of the circular segment \(A=R^2*\left(\arcsin{\frac{c}{2R}}-\frac{c}{2R}*\sqrt{1-(\frac{c}{2R})^2}\right)=R\left(\arccos\frac{d}{R}-d\sqrt{1-\frac{d^2}{R^2}}\right)\)

Do you know the derivation of the first formula:

\(\displaystyle c \ = \ 2 \ * \ R \ sin(\frac{\theta}{2})\)

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[Win_odd Dhamnekar]

Do you know the derivation of the first formula:

\(\displaystyle c \ = \ 2 \ * \ R \ sin(\frac{\theta}{2})\)

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

Hello,

I have understood all the 11 formulas with my simple logic. Last formulae for area of circular segment of a circle is wrongly stated in my original thread #1

It must be \(A=R^2\left(\arccos{\frac{d}{R}}-\frac{d}{R}\sqrt{1-\frac{d^2}{R^2}}\right)\) or \(R*\left(R\arccos{\frac{d}{R}}-d*\sqrt{1-\frac{d^2}{R^2}}\right)\) Thanks.