Sine of 1 Degree
Theorem
\(\ds \sin 1 \degrees = \sin \dfrac \pi {180}\) | \(=\) | \(\ds \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8} \paren {\sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {90 + 30 \sqrt 5} } } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8} \paren {\sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2\sqrt {10 + 2 \sqrt 5} + 2\sqrt {90 + 30 \sqrt 5} } } }\) |
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 $T$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {R + \sqrt {\paren {-\dfrac 1 4}^3 + R^2} } + \sqrt [3] {R - \sqrt {\paren {-\dfrac 1 4}^3 + R^2} }\) | substituting for $Q$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + \sqrt {\paren {-\dfrac 1 4}^3 + \dfrac {\sin^2 3 \degrees} {64} } } + \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 - \sqrt {\paren {-\dfrac 1 4}^3 + \dfrac {\sin^2 3 \degrees} {64} } }\) | substituting for $R$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + \sqrt {\dfrac {\sin^2 3 \degrees - 1} {64} } } + \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 - \sqrt {\dfrac {\sin^2 3 \degrees - 1} {64} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + \sqrt {\dfrac {-\cos^2 3 \degrees} {64} } } + \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 - \sqrt {\dfrac {-\cos^2 3 \degrees} {64} } }\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + i \dfrac {\cos 3 \degrees} 8 } + \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 - i \dfrac {\cos 3 \degrees} 8}\) | $i$ is a square root of $-1$, that is, $i = \sqrt {-1}$. | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] {-\dfrac {\sin 3 \degrees} 8 + i \dfrac {\cos 3 \degrees} 8 } + \sqrt [3] {-1 \paren {\dfrac {\sin 3 \degrees} 8 + i \dfrac {\cos 3 \degrees} 8} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \sqrt [3] {-\sin 3 \degrees + i \cos 3 \degrees } - \dfrac 1 2 \sqrt [3] {\sin 3 \degrees + i \cos 3 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \sqrt [3] {-\cos 87 \degrees + i \sin 87 \degrees } - \dfrac 1 2 \sqrt [3] {\cos 87 \degrees + i \sin 87 \degrees}\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \sqrt [3] {\cos 93 \degrees + i \sin 93 \degrees } - \dfrac 1 2 \sqrt [3] {\cos 87 \degrees + i \sin 87 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\sqrt [3] {\cis 93 \degrees} - \sqrt [3] {\cis 87 \degrees} }\) | KEY EQUATION |
- $\sqrt [3] {\cis 93 \degrees}$ has $3$ unique cube roots.
$\tuple {\cis 31 \degrees, \cis 151 \degrees, \cis 271 \degrees}$
- $\sqrt [3] {\cis 87 \degrees }$ also has $3$ unique cube roots.
$\tuple {\cis 29 \degrees, \cis 149 \degrees, \cis 269 \degrees}$
We need to investigate $9$ differences and find solutions where the complex part vanishes.
\(\ds 1) \ \ \) | \(\ds \dfrac 1 2 \paren {\sqrt [3] {\cis 93 \degrees } - \sqrt [3] {\cis 87 \degrees } }\) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 31 \degrees + i \sin 31 \degrees } - \paren {\cos 29 \degrees + i \sin 29 \degrees } }\) | No. $\paren {\sin 31 \degrees - \sin 29 \degrees } > 0$ | ||||||||||
\(\ds 2) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 31 \degrees + i \sin 31 \degrees } - \paren {\cos 149 \degrees + i \sin 149 \degrees } }\) | Yes. $\paren {\sin 31 \degrees - \sin 149 \degrees } = 0$ | ||||||||||
\(\ds 3) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 31 \degrees + i \sin 31 \degrees } - \paren {\cos 269 \degrees + i \sin 269 \degrees } }\) | No. $\paren {\sin 31 \degrees - \sin 269 \degrees } > 0$ | ||||||||||
\(\ds 4) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 151 \degrees + i \sin 151 \degrees } - \paren {\cos 29 \degrees + i \sin 29 \degrees } }\) | Yes. $\paren {\sin 151 \degrees - \sin 29 \degrees } = 0$ | ||||||||||
\(\ds 5) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 151 \degrees + i \sin 151 \degrees } - \paren {\cos 149 \degrees + i \sin 149 \degrees } }\) | No. $\paren {\sin 151 \degrees - \sin 149 \degrees } < 0$ | ||||||||||
\(\ds 6) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 151 \degrees + i \sin 151 \degrees } - \paren {\cos 269 \degrees + i \sin 269 \degrees } }\) | No. $\paren {\sin 151 \degrees - \sin 269 \degrees } > 0$ | ||||||||||
\(\ds 7) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 271 \degrees + i \sin 271 \degrees } - \paren {\cos 29 \degrees + i \sin 29 \degrees } }\) | No. $\paren {\sin 271 \degrees - \sin 29 \degrees } < 0$ | ||||||||||
\(\ds 8) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 271 \degrees + i \sin 271 \degrees } - \paren {\cos 149 \degrees + i \sin 149 \degrees } }\) | No. $\paren {\sin 271 \degrees - \sin 149 \degrees } < 0$ | ||||||||||
\(\ds 9) \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 271 \degrees + i \sin 271 \degrees } - \paren {\cos 269 \degrees + i \sin 269 \degrees } }\) | Yes. $\paren {\sin 271 \degrees - \sin 269 \degrees } = 0$ |
In solution $2)$ above, we rotate $\sqrt [3] {\cis 87 \degrees }$ by $120 \degrees$ (multiply by $\paren {-\dfrac 1 2 + i \dfrac {\sqrt 3} 2}$)
In solution $4)$ above, we rotate $\sqrt [3] {\cis 93 \degrees }$ by $120 \degrees$ (multiply by $\paren {-\dfrac 1 2 + i \dfrac {\sqrt 3} 2}$)
In solution $9)$ above, we would rotate BOTH $\sqrt [3] {\cis 93 \degrees}$ and $\sqrt [3] {\cis 87 \degrees}$ by $240 \degrees$ (multiply by $\paren {-\dfrac 1 2 - i \dfrac {\sqrt 3} 2}$)
For the remainder of this proof, we will arbitrarily choose solution $9)$:
\(\ds \sin 1 \degrees\) | \(=\) | \(\ds \paren { -\dfrac 1 2 - i \dfrac {\sqrt 3} 2 } \dfrac 1 2 \paren {\sqrt [3] {\cis 93 \degrees } - \sqrt [3] {\cis 87 \degrees } }\) | Solution $9)$ above - rotating by $240 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\cos 87 \degrees + i \sin 87 \degrees } - \sqrt [3] {\cos 87 \degrees + i \sin 87 \degrees } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\sin 3 \degrees + i \cos 3 \degrees } - \sqrt [3] {\sin 3 \degrees + i \cos 3 \degrees } }\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\sin 3 \degrees + i \sqrt { 1 - \sin^2 3 \degrees } } - \sqrt [3] {\sin 3 \degrees + i \sqrt {1 - \sin^2 3 \degrees } } }\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt { 1 - \paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} }^2 } } }\) | Sine of 3 Degrees | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren {\sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt { 1 - \paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} }^2 } } }\) |
Squaring the numerator of the Sine of 3 Degrees, we get:
\(\ds \paren {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} }^2\) | \(=\) | \(\ds 30 + 10\sqrt 3 - 6\sqrt 5 - 2\sqrt {15} - 6 \sqrt {50 + 10 \sqrt 5} + 2 \sqrt {150 + 30\sqrt 5}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 10\sqrt 3 + 10 - 2\sqrt {15} - 2\sqrt 5 - 2 \sqrt {150 + 30\sqrt 5} + 2 \sqrt {50 + 10 \sqrt 5}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 6\sqrt 5 - 2\sqrt {15} + 6 + 2\sqrt 3 + 6 \sqrt {10 + 2 \sqrt 5} - 2 \sqrt {30 + 6\sqrt 5}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 2\sqrt {15} - 2\sqrt 5 + 2\sqrt 3 + 2 + 2 \sqrt {30 + 6 \sqrt 5} - 2 \sqrt {10 + 2\sqrt 5}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 6 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {150 + 30\sqrt 5} + 6 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {30 + 6 \sqrt 5} + \paren {60 + 12 \sqrt 5 } - 4 \sqrt {90 + 30\sqrt 5}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \sqrt {150 + 30\sqrt 5} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {30 + 6\sqrt 5} - 2 \sqrt {10 + 2\sqrt 5} - 4 \sqrt {90 + 30\sqrt 5} + \paren {20 + 4 \sqrt 5 }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 128 + 24 \sqrt 3 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30\sqrt 5}\) | Aggregating terms |
Bringing that result back into our main result, we get:
\(\ds \) | \(=\) | \(\ds \paren {-\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt { 1 - \paren {\dfrac {128 + 24 \sqrt 3 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {256} } } } }\) | Sine of 3 Degrees | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\dfrac 1 4 - i \dfrac {\sqrt 3} 4} \paren { \sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt { 1 - \paren {\dfrac {128 + 24 \sqrt 3 - 8 \sqrt {15} - 8 \sqrt {50 + 10 \sqrt 5} + 8 \sqrt {10 + 2 \sqrt 5} - 8 \sqrt {90 + 30 \sqrt 5} } {256} } } } }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\dfrac 1 4 - i \dfrac {\sqrt 3} 4} \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt {\paren {\dfrac {128 - 24 \sqrt 3 + 8 \sqrt {15} + 8 \sqrt {50 + 10 \sqrt 5} - 8 \sqrt {10 + 2 \sqrt 5} + 8 \sqrt {90 + 30 \sqrt 5} } {256} } } } }\) | $\paren {1 - \dfrac {128} {256} } = \dfrac {128} {256}$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt {3} } 4 } \paren { \sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt {\paren {\dfrac {128 - 24 \sqrt 3 + 8 \sqrt {15} + 8 \sqrt {50 + 10 \sqrt 5} - 8 \sqrt {10 + 2 \sqrt 5} + 8 \sqrt {90 + 30\sqrt 5} } {256} } } } }\) |
Canceling $4$ from the numerator and denominator, we get:
\(\ds \) | \(=\) | \(\ds \paren { -\dfrac 1 4 - i \dfrac {\sqrt 3} 4 } \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt {\paren {\dfrac {32 - 6 \sqrt 3 + 2\sqrt {15} + 2\sqrt {50 + 10 \sqrt 5} - 2\sqrt {10 + 2 \sqrt 5} + 2\sqrt {90 + 30\sqrt 5} } {64} } } } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\dfrac 1 4 - i \dfrac {\sqrt 3} 4} \paren { \sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16} } + i \sqrt {\paren {\dfrac {32 - 6 \sqrt 3 + 2\sqrt {15} + 2\sqrt {50 + 10 \sqrt 5} - 2\sqrt {10 + 2 \sqrt 5} + 2\sqrt {90 + 30\sqrt 5} } {64} } } } }\) |
Moving an $8$ to the outside, we get:
\(\ds \) | \(=\) | \(\ds \paren {-\dfrac 1 8 - i \dfrac {\sqrt 3} 8} \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {90 + 30 \sqrt 5} } } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {-\dfrac 1 8 - i \dfrac {\sqrt 3} 8} \paren { \sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {90 + 30 \sqrt 5} } } }\) |
Flipping the top and bottom lines, we get our main result:
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8} \paren {\sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {90 + 30 \sqrt 5} } } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren { \dfrac 1 8 + i \dfrac {\sqrt {3} } 8 } \paren { \sqrt [3] {-\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2 \sqrt {10 + 2 \sqrt 5} + 2 \sqrt {90 + 30 \sqrt 5} } } }\) |
We can verify that this is correct by noting that:
\(\ds 8 \sin 3 \degrees\) | \(=\) | \(\ds \paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2}\) | ||||||||||||
\(\ds 8 \cos 3 \degrees\) | \(=\) | \(\ds \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqrt 5} - 2\sqrt {10 + 2 \sqrt 5} + 2\sqrt {90 + 30 \sqrt 5} }\) | ||||||||||||
\(\ds \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8}\) | \(=\) | \(\ds \dfrac 1 4 \cis 60 \degrees\) |
Plugging these into the main result, we see:
\(\ds \) | \(=\) | \(\ds \dfrac 1 4 \cis 60 \degrees \paren {\paren { \sqrt [3] {8 \sin 3 \degrees + i 8 \cos 3 \degrees } } - \paren { \sqrt [3] {-8 \sin 3 \degrees + i 8 \cos 3 \degrees } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \cis 60 \degrees \paren {\paren { \sqrt [3] {\sin 3 \degrees + i \cos 3 \degrees } } - \paren { \sqrt [3] {-\sin 3 \degrees + i \cos 3 \degrees } } }\) | moving the $8$ outside | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \cis 60 \degrees \paren {\paren { \sqrt [3] {\cos 87 \degrees + i \sin 87 \degrees } } - \paren { \sqrt [3] {\cos 93 \degrees + i \sin 93 \degrees } } }\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \cis 60 \degrees \paren { \cis 29 \degrees - \cis 31 \degrees }\) | Roots of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \cis 89 \degrees - \cis 91 \degrees }\) | Complex Multiplication as Geometrical Transformation | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { \paren {\cos 89 \degrees + i \sin 89 \degrees } - \paren {\cos 91 \degrees + i \sin 91 \degrees } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { 2 \paren {\cos 89 \degrees } }\) | Sine of Supplementary Angle and Cosine of Supplementary Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren { 2 \paren {\sin 1 \degrees } }\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin 1 \degrees\) |
$\blacksquare$