Evaluating sin(1)

[morson]

Problem: find the exact value of sin(1), in degrees.

This problem has been bugging me as of late. Constructing the correct polygon is very hard. Using de Moivre's formula leads me nowhere. Using identities doesn't yield anything either.

I have reached the equation:

\(\displaystyle \L\ 64y^3 - 48y + \sqrt{2}\(\sqrt{5}\ - 1)(\sqrt{3}\ + 1) + 2(1 - \sqrt{3}\)\sqrt{\sqrt{5}\ + 5}\ = 0\)

But solving this leads to a solution of sin(1).

Is there no way to obtain the exact value? Why have I been doing this? I am trying to write a table of particular integer values of sin(x). I have found all multiples of 3 (ie sin(3), sin(6), sin(9) ...), but if I can find sin(1), I can find the rest. Constructing a table of all sine values for all acute angles was a task set to us by my teacher.

 

[pka]

\(\displaystyle \L \sin (x) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{2k + 1} }}{{\left( {2k + 1} \right)!}}} \quad \Rightarrow \sin (1) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k }}{{\left( {2k + 1} \right)!}}}\)

 

[Deleted member 4993]

\(\displaystyle \L \sin (x) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{2k + 1} }}{{\left( {2k + 1} \right)!}}} \quad \Rightarrow \sin (1) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k }}{{\left( {2k + 1} \right)!}}}\)

In the equation above - x is in radian

 

[Opalg]

Problem: find the exact value of sin(1), in degrees.

Is there no way to obtain the exact value? Why have I been doing this? I am trying to write a table of particular integer values of sin(x). I have found all multiples of 3 (ie sin(3), sin(6), sin(9) ...), but if I can find sin(1), I can find the rest. Constructing a table of all sine values for all acute angles was a task set to us by my teacher.

There is a list of trig functions for multiples of 3 degrees at http://en.wikipedia.org/wiki/Exact_trig ... _constants

I believe that there is no exact formula for the sine of 1 degree.

 

[morson]

Then there is no exact solution to the cubic I posted?

 

[Opalg]

Then there is no exact solution to the cubic I posted?

I'm pretty sure that is correct: there is no closed-form solution to that cubic. But maybe someone else can confirm this.

 

[Deleted member 4993]

There is closed form solution to cubic. I believe it is called Caradn's formula or some such thing. A Google search can bring that out.

However, it is a horrendous expression....