Geometric Solution to the Sine of 36 Degrees
Okay folks, you have a neat challenge that awaits you on the geometric solution to the sine of 36 degrees. Please use this blog to post comments if you have questions. Remember, this will require you to use all your knowledge of geometry and don't be surprised if a quadratic equation pops up somewhere. You can use your calculators for the computations (to avoid spending hours doing arithmetic as the 10th century astronomers did) but only round off quantities at the very end. In fact, don't round off anything until you get your final answer. Work in your homework journals and show all your work.
Okay folks, you have a neat challenge that awaits you on the geometric solution to the sine of 36 degrees. Please use this blog to post comments if you have questions. Remember, this will require you to use all your knowledge of geometry and don't be surprised if a quadratic equation pops up somewhere. You can use your calculators for the computations (to avoid spending hours doing arithmetic as the 10th century astronomers did) but only round off quantities at the very end. In fact, don't round off anything until you get your final answer. Work in your homework journals and show all your work.Then, read the article on Islamic Astronomy by Owen Gingerich that I handed out at the end of class. This reading will provide you with all the necessary context for the work we are conducting.
(Now please do the reading or I'll be forced to give you another "reading quiz" and then let the dangling question of whether it'll be counted or not be much cause for emotional instability!)
(Now please do the reading or I'll be forced to give you another "reading quiz" and then let the dangling question of whether it'll be counted or not be much cause for emotional instability!)
14 comments:
I think I found Y, but I am not sure how to find sin and cos after that. Help?
If you've found Y, then think of what other lengths are equal to Y. Where else is the angle alpha occurring on the diagram? is AD also equal to Y?
Make sure you have determined which length you are ultimately trying to find. Which length represents the sine of 36 degrees?
I think sine of 36 degrees is BF.
im having trouble finding y. can anyone help me? i proved the triangles are similar but i dont know what to do after that.
Victoria, I agree with you!
Virginia, can you find the measures of the angle sigma and the angle epsilon (epsilon is the E looking symbol and sigma is the snake looking symbol, a bit like an S)?
yes i think i found those, but i'm not sure what to do with them.
I'm getting stuck on the second part of (c). I have gotten all the way there but then just may need clearing up on directions or actual help. Where is a good place to start with the equation comparison wise?
I've been having the same problem as Vranian, I've tried leaving the problem alone for a while then coming back, but it has not helped. Any ideas?
that's where i'm stuck too. i'm not sure how you can use similar triangles to find y.
If you've done your calculations correctly and know the value of epsilon and sigma then you shoudl be able to see another isosceles triangle and this should tell you something about the length DC.
sin 36 = 2 sin 18 cos 18
= 2 sin 18 sin 72
= 2 sin 18 * (2 sin 36 cos 36)
= sin 36 * 4 sin 18 cos 36
1/4 = sin 18 cos 36
= sin 18 sin 54
= sin 18 (sin 18 cos 36 + cos 18 sin 36)
= sin 18 (sin 18 (1 - 2 sin^2 18) + cos 18 (2 sin 18 cos 18))
= sin 18 (sin 18 (1 - 2 sin^2 18) + sin 18 (2 cos^2 18))
= sin^2 18 (1 + 2 cos^2 18 - 2 sin^2 18)
= sin^2 18 (1 + 2 cos 36)
= sin^2 18 (1 + 2 sin 54)
= sin^2 18 + 2 sin^2 18 sin 54
= sin^2 18 + 2 sin 18 (sin 18 sin 54)
= sin^2 18 + 2 sin 18 * 1/4
= sin^2 18 + (sin 18)/2
sin^2 18 + 1/2 * sin 18 - 1/4 = 0
sin 18 = (-1 + sqrt(5))/4
cos 36 = 1 - 2 sin^2 18 = 1 - (3 - sqrt(5))/4 = (1 + sqrt(5))/4
cos^2 36 = (6 + 2 sqrt(5)) / 16 = (3 + sqrt(5)) / 8
sin^2 36 = 1 - cos^2 36 = (5 - sqrt(5)) / 8
sin 36 = sqrt((5 - sqrt(5)) / 8) = 1/2 * sqrt((5 - sqrt(5)) / 2)
thats a way my friend showed me to calculate the value of sin(36)... I could not find a scientific calculator before, but now I have one, so it's all good
And how do you know the sine of 18 degrees? And then why even bother with the sine of 18 degrees if you are going to use your calculator. In fact, you could have just typed the sine of 36 degrees in your calculator and wala, you would've had it. Silly, silly notion....
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