{{begin-eqn}} {{eqn | l = \sin \left({3 \times 1^\circ}\right)       | r = 3 \sin 1^\circ - 4 \sin^3 1^\circ       | c = Triple Angle Formula for Sine }} {{eqn | l = \sin 3^\circ       | r = 3 \sin 1^\circ - 4 \sin^3 1^\circ }} {{eqn | l = 4 \sin^3 1^\circ - 3 \sin 1^\circ + \sin 3^\circ       | r = 0 }} {{end-eqn}} This is in the form: :$a x^3 + b x^2 + c x + d = 0$ where: : $x = \sin 1^\circ$ : $a = 4$ : $b = 0$ : $c = -3$ : $d = \sin 3^\circ$ From Cardano's Formula: :$x = S + T$ where: :$S = \sqrt [3] {R + \sqrt{Q^3 + R^2}}$ :$T = \sqrt [3] {R - \sqrt{Q^3 + R^2}}$ where: {{begin-eqn}} {{eqn | l = Q       | r = \dfrac {3 a c - b^2} {9 a^2}       | c =  }} {{eqn | r = \dfrac {3 \times 4 \times \left({-3}\right) - 0^2} {9 \times 4^2}       | c =  }} {{eqn | r = -\dfrac 1 4       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = R       | r = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}       | c =  }} {{eqn | r = \dfrac {9 \times 4 \times 0 \times \left({-3}\right) - 27 \times 4^2 \times \sin 3^\circ - 2 \times 0^3} {54 \times 4^3}       | c =  }} {{eqn | r = \dfrac {\sin 3^\circ} 8       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \sin 1^\circ       | r = S + T       | c = putting $x = S + T$ }} {{eqn | r = \sqrt[3] {R + \sqrt{Q^3 + R^2} } + \sqrt[3] {R - \sqrt{Q^3 + R^2} }       | c = substituting for $S$ and $T$ }} {{eqn | r = \sqrt[3] {R + \sqrt{-\left({\dfrac 4 9}\right)^3 + R^2} } + \sqrt[3] {R - \sqrt {-\left({\dfrac 4 9}\right)^3 + R^2} }       | c = substituting for $Q$ }} {{eqn | r = \sqrt[3] {-\dfrac{\sin 3^\circ} 8 + \sqrt{-\left({ \dfrac 4 9}\right)^3 + \dfrac {\sin^2 3^\circ} {64} } }       | c = substituting for $R$ }} {{eqn | o =        | ro= +       | r = \sqrt[3] {-\dfrac {\sin 3^\circ} 8 - \sqrt{-\left({ \dfrac 4 9}\right)^3 + \dfrac {\sin^2 3^\circ} {64} } }       | c =  }} {{end-eqn}} {{mistake|I believe the substituted value for $Q$ above may be wrong. It appears that $Q {{=}} -\dfrac 4 9$ has been used, but in the calculation above it may be the case that it should be $Q {{=}} -\dfrac 1 4$. The algebra has gone wrong somewhere and needs to be reviewed.}} By Sine of 3 Degrees: {{begin-eqn}} {{eqn | l = \sin 1^\circ       | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} + \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right)^2 }{64} } } }} {{eqn | o =        | ro= +       | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} - \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{\left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right)^2}{64} } } }} {{eqn | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} + \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{ \left({ \dfrac{-8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16^2} }\right) }{64} } } }} {{eqn | o =        | ro= +       | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} - \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{\left({ \dfrac{-8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16^2} }\right)}{64} } } }} {{eqn | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} + \sqrt{-\dfrac{64}{729} + \dfrac{ -8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16384} } } }} {{eqn | o =        | ro= +       | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} - \sqrt{-\dfrac{64}{729} + \dfrac{ -8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16384} } } }} {{eqn | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} + \sqrt{-\dfrac{64}{729} + \dfrac{ -1 + 3\sqrt3 - 2\sqrt5 - \sqrt{15} - \sqrt {50 + 10\sqrt5} + \sqrt {10 + 2 \sqrt5} - \sqrt{90 + 30\sqrt5} }{2048} } } }} {{eqn | o =        | ro= +       | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} - \sqrt{-\dfrac{64}{729} + \dfrac{ -1 + 3\sqrt3 - 2\sqrt5 - \sqrt{15} - \sqrt {50 + 10\sqrt5} + \sqrt {10 + 2 \sqrt5} - \sqrt{90 + 30\sqrt5} }{2048} } } }} {{end-eqn}} {{qed}} Category:Sine Function hhuqg80tw9ly32lbnlu5xcnr2qlgnct