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

1. Just set , expand out and you have a polynomial. I can't be bothered right now but I guess that's what's been done.
2. (Original post by Ano123)
I don't know if there is a quick way of obtaining these answers for all of the 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.
3. (Original post by morgan8002)
Just set , 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 .
4. I was intrigued because finding the values of was easy and the answers were very nice.
This was the next set of angles to try obviously.
5. (Original post by Ano123)
Yeah basically, except .
Oh, yeah. The general formula for the hexic follows from there, using different values for . The solution would be harder to find generally I think, since different radicals in the cubic formula might be positive or negative.
6. (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.
7. (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.
8. (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
9. (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.
10. (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.
11. (Original post by Student403)
That and also this isn't in further maths xD
The tools are. It's not a big leap.
12. (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.
13. (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
14. (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.
Aye, I'll have to agree with you there. Maths is awesome.
15. (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.
16. (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.
Aye, was this back in Cardano's time?
17. (Original post by Zacken)
Aye, was this back in Cardano's time?
Yes, when I was a young man.
18. (Original post by Gregorius)
Yes, when I was a young man.

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