October 25th, 2011, 07:58 AM  #1 
Senior Member Joined: Sep 2011 Posts: 140 Thanks: 0  Surds
We know that sin30 is 1/2 and sin45= 1/sqare root 2 etc... these angles in trigo. forms have a exact value, as in even though sin45 is 0.707106781...... we can still express it in terms of surds like; . However there are some random numbers like 0.787186896183961.... that cannot be expressed in terms of surds like sin45. Now my question is; can all angles in trigonometric forms be expressed exactly for all real numbers, not complex?? Like for example sin5; can be expressed exactly?? or any angle. 
October 25th, 2011, 09:58 AM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Surds
Check this out... it might be a little heavy, but this is the direction you'll need to go to approach such questions. http://en.wikipedia.org/wiki/Transcende ... scendental 
October 25th, 2011, 05:02 PM  #3 
Senior Member Joined: Sep 2011 Posts: 140 Thanks: 0  Re: Surds
The site does state one statement I am looking for; sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem). However, it doesn't tell me whether it's real or complex. I am a little confused... Can anyone clarify? Thank you. 
October 25th, 2011, 11:50 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
For a complex nonzero algebraic number a, it's still the case that sin(a) is transcendental (real or complex).


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