Determinant

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#1 h Apr 16, 2025, 3:01 AM

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Calculate the determinant:
$\begin{vmatrix} 1 & \sin\alpha & \sin2\alpha & \sin3\alpha\\ \sin\alpha & \sin2\alpha & \sin3\alpha & \sin4\alpha\\ \sin2\alpha & \sin3\alpha & \sin4\alpha & \sin5\alpha\\ \sin3\alpha & \sin4\alpha & \sin5\alpha & \sin6\alpha\\ \end{vmatrix}$

This post has been edited 1 time. Last edited by Saucepan_man02, Apr 16, 2025, 3:02 AM
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#2 h Apr 16, 2025, 9:19 AM

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$D=\left|\begin{array}{llll} 1 & \sin \alpha & \sin 2\alpha & \sin 3\alpha \\ \sin \alpha & \sin 2\alpha & \sin 3\alpha & \sin 4\alpha \\ \sin 2\alpha & \sin 3\alpha & \sin 4\alpha & \sin 5\alpha \\ \sin 3\alpha & \sin 4\alpha & \sin 5\alpha & \sin 6\alpha \end{array}\right|$ .

$D\stackrel{C_{4}+C_{2}}{=}\left|\begin{array}{llll} 1 & \sin \alpha & \sin 2\alpha & \sin 3\alpha +\sin \alpha\\ \sin \alpha & \sin 2\alpha & \sin 3\alpha & \sin 4\alpha +\sin 2\alpha \\ \sin 2\alpha & \sin 3\alpha & \sin 4\alpha & \sin 5\alpha +\sin 3\alpha \\ \sin 3\alpha & \sin 4\alpha & \sin 5\alpha & \sin 6\alpha +\sin 4\alpha\end{array}\right|$ .

$D=\left|\begin{array}{llll} 1 & \sin \alpha & \sin 2\alpha & 2\sin 2\alpha \cos \alpha\\ \sin \alpha & \sin 2\alpha & \sin 3\alpha & 2\sin 3\alpha \cos \alpha \\ \sin 2\alpha & \sin 3\alpha & \sin 4\alpha & 2\sin 4\alpha \cos \alpha \\ \sin 3\alpha & \sin 4\alpha & \sin 5\alpha & 2\sin 5\alpha \cos \alpha\end{array}\right|\ =\ 0$ .

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