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 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; $\frac{1}{\sqrt{2}}$. 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 non-zero 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: 17,160 Thanks: 1284 For a complex non-zero algebraic number a, it's still the case that sin(a) is transcendental (real or complex).

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