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Math Help - Sinus product

  1. #1
    Super Member dhiab's Avatar
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    Sinus product

    Calculate : p = \sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times ..... \times \sin 89^\circ \times \sin 90^\circ <br />
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by dhiab View Post
    Calculate : p = \sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times ..... \times \sin 89^\circ \times \sin 90^\circ <br />
    Outline solution: For x = 1,2,3,...,90, \sin(180x)=0 (where x is measured in degrees). The formula for \sin(180x) is of the form \sin(180x) = \cos x\sin x(180 - \ldots - 2^{179}\sin^{178}x). The part in parentheses is a polynomial of degree 178 in \sin x, with all but the first and last terms omitted.

    If we put that expression equal to 0 then the factor \cos x=0 corresponds to x=90 and x=90, and the factor \sin x=0 corresponds to x=0. If we write s for \sin x then the 178 roots of the polynomial 180 - \ldots - 2^{179}s^{178} are \sin x for x=\pm1,\pm2,\ldots,\pm89. But the product of the roots of a polynomial of even degree is the constant term divided by the coefficient of the leading term. Therefore -\sin^21^\circ\sin^22^\circ\cdots\sin^289^\circ = \frac{-180}{2^{179}}, and hence \sin1^\circ\sin2^\circ\cdots\sin89^\circ = \frac{\sqrt{90}}{2^{89}} (and of course \sin90^\circ=1, so that does not contribute to the product).
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