MHB Is There a Rational Multiple for the Product of Sines from 1 to 90 Degrees?

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The discussion centers on proving the existence of a rational number k such that the product of sines from 1 to 90 degrees equals k multiplied by the square root of 10. Participants share their solutions, with Olinguito and Opalg providing correct answers. The thread emphasizes the importance of following the guidelines for submitting solutions. The problem highlights interesting properties of trigonometric functions and their products. Overall, the discussion fosters engagement with mathematical problem-solving.
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Here is this week's POTW:

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Show that there is a rational number $k$ such that $\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ \sin 90^\circ =k\sqrt{10}$.

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Congratulations to the following members for their correct solution!(Cool)

1. Olinguito
2. Opalg

Solution from Olinguito:
Observe that
$$2\sin x^\circ\sin(60-x)^\circ$$
$=\ \cos(2x-60)^\circ-\cos60^\circ$

$=\ \cos(2x-60)^\circ-\dfrac12$

and so
$$4\sin x^\circ\sin(60-x)^\circ\sin(60+x)^\circ$$
$=\ 2\left[\cos(2x-60)^\circ-\dfrac12\right]\sin(60+x)^\circ$

$=\ 2\cos(2x-60)^\circ\sin(60+x)^\circ-\sin(60+x)^\circ$

$=\ \sin(3x)^\circ+\sin(120-x)^\circ-\sin(60+x)^\circ$

$=\ \sin(3x)^\circ+\sin(180-[120-x])^\circ-\sin(60+x)^\circ$

i.e.
$$\boxed{\sin x^\circ\sin(60-x)^\circ\sin(60+x)^\circ\ =\ \frac{\sin(3x)^\circ}4}.$$

So:
$$\sin1^\circ\sin2^\circ\cdots\sin90^\circ$$
$=\ (\sin1^\circ\sin59^\circ\sin61^\circ)(\sin2^\circ\sin58^\circ\sin62^\circ)\cdots(\sin29^\circ\sin31^\circ\sin89^\circ)\sin30^\circ\sin60^\circ\sin90^\circ$

$=\ \dfrac{\sin3^\circ}4\cdot\dfrac{\sin6^\circ}4\cdots\dfrac{\sin87^\circ}4\cdot\dfrac12\cdot\dfrac{\sqrt3}2\cdot1$

$=\ \dfrac{\sqrt3}{4^{30}}\sin3^\circ\sin6^\circ\cdots\sin87^\circ$

$=\ \dfrac{\sqrt3}{4^{30}}(\sin3^\circ\sin57^\circ\sin63^\circ)(\sin6^\circ\sin54^\circ\sin66^\circ)\cdots(\sin27^\circ\sin33^\circ\sin87^\circ)\sin30^\circ\sin60^\circ$

$=\ \dfrac{\sqrt3}{4^{30}}\cdot\dfrac{\sin9^\circ}4\cdot\dfrac{\sin18^\circ}4\cdots\dfrac{\sin81^\circ}4\cdot\dfrac12\cdot\dfrac{\sqrt3}2$

$=\ \dfrac3{4^{40}}\sin9^\circ\sin18^\circ\cdots\sin81^\circ$

$=\ \dfrac3{4^{40}}(\sin9^\circ\sin81^\circ)(\sin18^\circ\sin72^\circ)(\sin27^\circ\sin63^\circ)(\sin36^\circ\sin54^\circ)\sin45^\circ$

$=\ \dfrac3{4^{40}}(\sin9^\circ\cos9^\circ)(\sin18^\circ\cos18^\circ)(\sin27^\circ\cos27^\circ)(\sin36^\circ\cos36^\circ)\cdot\dfrac{\sqrt2}2$

$=\ \dfrac{3\sqrt2}{2^{81}}\cdot\dfrac{\sin18^\circ}2\dfrac{\sin36^\circ}2\cdot\dfrac{\sin54^\circ}2\cdot\dfrac{\sin72^\circ}2$

$=\ \dfrac{3\sqrt2}{2^{85}}\sin18^\circ\cos18^\circ\sin36\cos36^\circ$

$=\ \dfrac{3\sqrt2}{2^{87}}\sin36^\circ\sin72^\circ$

$=\ \dfrac{3\sqrt2}{2^{86}}\sin^236^\circ\cos36^\circ$

$=\ \dfrac{3\sqrt2}{2^{86}}(\cos36^\circ-\cos^336^\circ).$

It remains to work out $\cos36^\circ$. We have
$$\sin36^\circ\ =\ \cos54^\circ$$
$\implies\ 2\sin18^\circ\cos18^\circ\ =\ 4\cos^318^\circ-3\cos18^\circ$

$\implies\ 2\sin18^\circ\ =\ 4\cos^218^\circ-3\ =\ 4-4\sin^218^\circ-3\ =\ 1-4\sin^218^\circ$

$\implies\ 4\sin^218^\circ+2\sin18^\circ-1\ =\ 0$

$\implies\ \sin18^\circ\ (>0) =\ \dfrac{-2+\sqrt{20}}8\ =\ \dfrac{-1+\sqrt5}4.$

So $\cos36^\circ\ =\ 1-2\sin^218^\circ\ =\ \dfrac{1+\sqrt5}4$

$\implies\ \cos^336^\circ\ =\ \dfrac{2+\sqrt5}8$

$\implies \cos36^\circ-\cos^336^\circ\ =\ \dfrac{\sqrt5}8$.

Hence
$$\sin1^\circ\sin2^\circ\cdots\sin90^\circ$$
$=\ \dfrac{3\sqrt2}{2^{86}}\cdot\dfrac{\sqrt5}8\ =\ \boxed{k\sqrt{10}\ \text{where}\ k=\dfrac3{2^{89}}\in\mathbb Q}$.


Alternate solution from Opalg:
This proof uses Chebyshev polynomials. By equation (14) in Multiple-Angle Formulas -- from Wolfram MathWorld, $\sin(180\theta) = -\cos\theta\, U_{179}(\sin \theta)$, where $U_{179}(x)$ is a Chebyshev polynomial of the second kind. And by equation (17) in Chebyshev Polynomial of the Second Kind -- from Wolfram MathWorld, $$U_{179}(x) = \sum_0^{89}(-1)^n{179-n\choose n}(2x)^{179-2n}.$$ It follows that $$\sin(180\theta) = -\cos\theta \sum_0^{89}(-1)^n{179-n\choose n}(2\sin\theta)^{179-2n}. \qquad(*)$$ For every integer $k$, $\sin(180k^\circ) = 0$. If $-89\leqslant k \leqslant 89$ it is also true that $\cos k^\circ \ne0$. It follows from (*) that for $-89\leqslant k \leqslant 89$, $\sin k^\circ$ satisfies the equation $$ \sum_0^{89}(-1)^n{179-n\choose n}(2x)^{179-2n} = 0. \qquad(**)$$ But that is an equation of degree 179, and the 179 solutions $\sin k^\circ \ (-89\leqslant k \leqslant 89)$ are all distinct. So they comprise all the solutions. One of the solutions is $x=0$ (corresponding to $k=0$), and the others are $\pm\sin k^\circ \ (1\leqslant k \leqslant 89)$ (because $\sin(-\theta) = -\sin\theta$). Divide (**) by $x$ to eliminate the solution $x=0$, and it follows that the solutions of $$ \sum_0^{89}(-1)^n{179-n\choose n}2^{179-2n}x^{178-2n} = 0$$ are $\pm\sin k^\circ \ (1\leqslant k \leqslant 89)$. Write the equation as $$ 2^{179}x^{178} - \ldots - 2{90\choose 89} = 0, \\ 2^{178}x^{178} - \ldots - 90 = 0, $$ to see that the product of the roots is $$(\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ)^2 = \dfrac{90}{2^{178}}.$$ Now take the square root and toss in the fact that $\sin90^\circ = 1$, to get $$\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ \sin 90^\circ = \frac3{2^{89}}\sqrt{10}.$$
 
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