I have the exact values for the sine of integers. Has this been accomplished before? Jim Parent
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In radians, the sine of any nonzero integer angle is a transcendental number. This follows from Lindemann–Weierstrass theorem. As far as I know, there is no closed-form expression for $\sin(1)$ that doesn't somehow involve transcendental functions Using degrees, the sine of any integer is an algebraic number, i.e. a root of a polynomial with integer coefficients. For $\sin(1^\circ)$, the smallest such polynomial has degree 48. It is also possible to express $\sin(1^\circ)$ by radicals as follows: $$ \sin(1^\circ) \;=\; \frac{(\sqrt[180]{-1})^{89} - (\sqrt[180]{-1})^{91}}{2} $$ where $\sqrt[180]{-1}$ denotes the principal 180'th root of $-1$, i.e. $\sqrt[180]{-1} = \cos(1^\circ) + i \sin(1^\circ)$. More generally, $$ \sin(k^\circ) \;=\; \frac{(\sqrt[180]{-1})^{90-k} - (\sqrt[180]{-1})^{90+k}}{2} $$ for any integer $k$. |
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