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Trigonometric functions - exact values ??!!??£@&&@?!

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Just set sin7x=1\sin 7x = 1, expand out and you have a polynomial. I can't be bothered right now but I guess that's what's been done.
Original post by Ano123
I don't know if there is a quick way of obtaining these answers for all of the sin(kπ7) \displaystyle \sin \left ( \frac{k\pi}{7} \right ) values but I just formed the equation above, and solved it using the cubic formula.
If there is a quicker way of doing these (maybe a general formula or something similar) please let me know.


Start thinking about roots of unity and cyclotomic polynomials.
Reply 22
Original post by morgan8002
Just set sin7x=1\sin 7x = 1, expand out and you have a polynomial. I can't be bothered right now but I guess that's what's been done.


Yeah basically, except sin7x=0 \sin{7x}=0 .
Reply 23
I was intrigued because finding the values of sin(kπ5) \sin \left ( \frac{k\pi}{5} \right ) was easy and the answers were very nice.
This was the next set of angles to try obviously.
Original post by Ano123
Yeah basically, except sin7x=0 \sin{7x}=0 .


Oh, yeah. The general formula for the hexic follows from there, using different values for sin7x\sin 7x. The solution would be harder to find generally I think, since different radicals in the cubic formula might be positive or negative.
Original post by Ano123
I find it weird that you can't get rid of the imaginary terms in this exact answer, but the answer itself is purely real.


Funnily enough, it was this observation in the context of the solution of cubic and quartic polynomial equations by radicals that led to the acceptance of the notion of complex numbers.
Reply 26
Original post by C-rated
This is the kind of crap that made me take physics instead of further maths


This makes zero sense, physics is full of ugly answers and approximations whereas mathematics is usually full of elegant and neat proofs.

Original post by Ano123
I find it weird that you can't get rid of the imaginary terms in this exact answer, but the answer itself is purely real.


Uh, why is that weird? Casus irreducibilis.
Original post by Zacken
This makes zero sense, physics is full of ugly answers and approximations whereas mathematics is usually full of elegant and neat proofs.



Uh, why is that weird? Casus irreducibilis.


That and also this isn't in further maths xD
Reply 28
Original post by Zacken
This makes zero sense, physics is full of ugly answers and approximations whereas mathematics is usually full of elegant and neat proofs.


Uh, why is that weird? Casus irreducibilis.

I think it is not very intuitive at all.
Reply 29
Original post by B_9710
I think it is not very intuitive at all.


It makes sense right away that square roots are messy enough such that it something is stuck in there, it's hard to turn it into something else (pardon the informality) but that's how I felt about it.
Original post by Student403
That and also this isn't in further maths xD


The tools are. It's not a big leap.
Reply 31
Original post by Zacken
It makes sense right away that square roots are messy enough such that it something is stuck in there, it's hard to turn it into something else (pardon the informality) but that's how I felt about it.


I thought about the square root analogy but even still, I think it's amazing that some real things can only be expressed exactly using complex terms. That's maths for you isn't it. :smile:
Original post by morgan8002
The tools are. It's not a big leap.


I assumed C-rated made the comment purely because of the way the root looked - which is what I was referring to in my comment on FM
Reply 33
Original post by B_9710
I thought about the square root analogy but even still, I think it's amazing that some real things can only be expressed exactly using complex terms. That's maths for you isn't it. :smile:


Aye, I'll have to agree with you there. Maths is awesome. :h:
Original post by Zacken

Uh, why is that weird? Casus irreducibilis.


It's not weird to us, as we are so used to the idea! At the time this was realized it was a major intellectual leap forward. :smile:
Reply 35
Original post by Gregorius
It's not weird to us, as we are so used to the idea! At the time this was realized it was a major intellectual leap forward. :smile:


Aye, was this back in Cardano's time?
Original post by Zacken
Aye, was this back in Cardano's time?


Yes, when I was a young man. :u:
Reply 37
Original post by Gregorius
Yes, when I was a young man. :u:


:rofl:

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