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Math Help - Exact values of sin20 and sin40 - impossible?

  1. #1
    RAz
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    Exact values of sin20 and sin40 - impossible?

    I'm trying to find exact trigonometric values. Say I have sin11, I would use the special ratios:

    sin11=sin(45-34)= etc. etc. and I would arrive with a surd answer, which is fine.

    But I'm having trouble with angles like 20 and 40. Is there a reason for this?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by RAz View Post
    I'm trying to find exact trigonometric values. Say I have sin11, I would use the special ratios:

    sin11=sin(45-34)= etc. etc. and I would arrive with a surd answer, which is fine.

    But I'm having trouble with angles like 20 and 40. Is there a reason for this?
    Most of the are difficult to find.

    For example \sin(20^{\circ})=\sin\left(\frac{\pi}{9}\right). One can use this to get \frac{\sqrt{3}}{2}\sin\left(\frac{\pi}{3}\right)=\  sin\left(\frac{\pi}{9}\cdot 3\right)=3\sin\left(\frac{\pi}{9}\right)-4\sin^3\left(\frac{\pi}{9}\right).

    Thus, \sin\left(\frac{\pi}{9}\right) is the solution to the depressed cubic x^3+px=q....I did something like this somewhere, hold on....ah! look here. That'll give you a taste of what solving them is like.
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    Senior Member Stroodle's Avatar
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