Sine of 1 Degree

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Theorem

\(\ds \sin 1 \degrees = \sin \dfrac {\pi} {180}\) \(=\) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {128} + \sqrt {-\dfrac {64} {729} + \dfrac {-1 + 3 \sqrt 3 - 2 \sqrt 5 - \sqrt {15} - \sqrt {50 + 10 \sqrt 5} + \sqrt {10 + 2 \sqrt 5} - \sqrt {90 + 30 \sqrt 5} } {2048} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {128} - \sqrt {-\dfrac {64} {729} + \dfrac {-1 + 3 \sqrt 3 - 2 \sqrt 5 - \sqrt {15} - \sqrt {50 + 10 \sqrt 5} + \sqrt {10 + 2 \sqrt 5} - \sqrt {90 + 30 \sqrt 5} } {2048} } }\)

where $\sin$ denotes the sine function.


Proof

\(\ds \map \sin {3 \times 1 \degrees}\) \(=\) \(\ds 3 \sin 1 \degrees - 4 \sin^3 1 \degrees\) Triple Angle Formula for Sine
\(\ds \sin 3 \degrees\) \(=\) \(\ds 3 \sin 1 \degrees - 4 \sin^3 1 \degrees\)
\(\ds 4 \sin^3 1 \degrees - 3 \sin 1 \degrees + \sin 3 \degrees\) \(=\) \(\ds 0\)


This is in the form:

$a x^3 + b x^2 + c x + d = 0$

where:

$x = \sin 1 \degrees$
$a = 4$
$b = 0$
$c = -3$
$d = \sin 3 \degrees$


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:

\(\ds Q\) \(=\) \(\ds \dfrac {3 a c - b^2} {9 a^2}\)
\(\ds \) \(=\) \(\ds \dfrac {3 \times 4 \times \paren {-3} - 0^2} {9 \times 4^2}\)
\(\ds \) \(=\) \(\ds -\dfrac 1 4\)

and:

\(\ds R\) \(=\) \(\ds \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}\)
\(\ds \) \(=\) \(\ds \dfrac {9 \times 4 \times 0 \times \paren {-3} - 27 \times 4^2 \times \sin 3 \degrees - 2 \times 0^3} {54 \times 4^3}\)
\(\ds \) \(=\) \(\ds \dfrac {\sin 3 \degrees} 8\)


Thus:

\(\ds \sin 1 \degrees\) \(=\) \(\ds S + T\) putting $x = S + T$
\(\ds \) \(=\) \(\ds \sqrt [3] {R + \sqrt {Q^3 + R^2} } + \sqrt[3] {R - \sqrt {Q^3 + R^2} }\) substituting for $S$ and $T$
\(\ds \) \(=\) \(\ds \sqrt [3] {R + \sqrt {-\paren {\dfrac 4 9}^3 + R^2} } + \sqrt [3] {R - \sqrt {-\paren {\dfrac 4 9}^3 + R^2} }\) substituting for $Q$
\(\ds \) \(=\) \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\sin^2 3 \degrees} {64} } }\) substituting for $R$
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 - \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\sin^2 3 \degrees} {64} } }\)



By Sine of 3 Degrees:

\(\ds \sin 1 \degrees\) \(=\) \(\ds \sqrt [3] {-\dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } } 8 + \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} }^2} {64} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } } 8 - \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} }^2} {64} } }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {-\dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } } 8 + \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\paren {\dfrac {-8 + 24 \sqrt 3 - 16 \sqrt 5 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {16^2} } } {64} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } } 8 - \sqrt {-\paren {\dfrac 4 9}^3 + \dfrac {\paren {\dfrac {-8 + 24 \sqrt 3 - 16 \sqrt 5 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {16^2} } } {64} } }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5 } + 2 \sqrt {5 + \sqrt 5} } {128} + \sqrt {-\dfrac {64} {729} + \dfrac {-8 + 24 \sqrt 3 - 16 \sqrt 5 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {16384} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {128} - \sqrt {-\dfrac {64} {729} + \dfrac {-8 + 24 \sqrt 3 - 16 \sqrt 5 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {16384} } }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {128} + \sqrt {-\dfrac {64} {729} + \dfrac {-1 + 3 \sqrt 3 - 2 \sqrt 5 - \sqrt {15} - \sqrt {50 + 10 \sqrt 5} + \sqrt {10 + 2 \sqrt 5} - \sqrt {90 + 30 \sqrt 5} } {2048} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sqrt [3] {-\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {128} - \sqrt {-\dfrac {64} {729} + \dfrac {-1 + 3 \sqrt 3 - 2 \sqrt 5 - \sqrt {15} - \sqrt {50 + 10 \sqrt 5} + \sqrt {10 + 2 \sqrt 5} - \sqrt {90 + 30 \sqrt 5} } {2048} } }\)

$\blacksquare$