De exacte waarde van sin(36°)

$\eqalign{ & \sin (36^\circ ) = \frac{{\sqrt {10 - 2\sqrt 5 } }} {4} \cr & \sin (36^\circ ) \cr & \downarrow x = 36^\circ \cr & 5x = 180^\circ \cr & 2x + 3x = 180^\circ \cr & \downarrow \sin \left( \alpha \right) = \sin (180^\circ - \alpha ) \cr & \sin (2x) = \sin (3x) \cr & 2\sin (x)\cos (x) = 3\sin (x) - 4\sin ^3 (x) \cr & \downarrow \cos (x) = \sqrt {1 - \sin ^2 (x)} \cr & 2\sin (x)\sqrt {1 - \sin ^2 (x)} = 3\sin (x) - 4\sin ^3 (x) \cr & \downarrow u = \sin (x) \cr & 2u\sqrt {1 - u^2 } = 3u - 4u^3 \cr & \left( {2u\sqrt {1 - u^2 } } \right)^2 = \left( {3u - 4u^3 } \right)^2 \cr & 4u^2 \left( {1 - u^2 } \right) = 9u^2 - 24u^4 + 16u^6 \cr & 4u^2 - 4u{}^4 = 9u^2 - 24u^4 + 16u^6 \cr & 16u^6 - 20u^4 + 5u^2 = 0 \cr & u^2 (16u^4 - 20u^2 + 5) = 0 \cr & u^2 = 0\,\,(v.n.) \vee 16u^4 - 20u^2 + 5 = 0 \cr & 16u^4 - 20u^2 + 5 = 0 \cr & \frac{1} {4}\left( {64u^4 - 80u^2 } \right) + 5 = 0 \cr & \frac{1} {4}\left( {\left( {8u^2 - 5} \right)^2 - 25} \right) + 5 = 0 \cr & \left( {\left( {8u^2 - 5} \right)^2 - 25} \right) + 20 = 0 \cr & \left( {8u^2 - 5} \right)^2 = 5 \cr & 8u^2 - 5 = \pm \sqrt 5 \cr & 8u^2 = 5 \pm \sqrt 5 \cr & u^2 = \frac{{5 \pm \sqrt 5 }} {8} \cr & u^2 = \frac{{5 - \sqrt 5 }} {8} \vee u^2 = \frac{{5 + \sqrt 5 }} {8}\,\,\,(v.n.) \cr & u = - \sqrt {\frac{{5 - \sqrt 5 }} {8}} \,\,(v.n.) \vee u = \sqrt {\frac{{5 - \sqrt 5 }} {8}} \cr & u = \sqrt {\frac{{5 - \sqrt 5 }} {8}} \cr & u = \sqrt {\frac{{10 - 2\sqrt 5 }} {{16}}} \cr & u = \frac{{\sqrt {10 - 2\sqrt 5 } }} {4} \cr}$

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