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Using the light-cone sum rule technique, we calculate the $B\\to K_{1A} (1^3P_1)$ and $B\\to K_{1B} (1^1P_1)$ tensor form factors, $T_1^{K_{1A}}(0)$ and $T_1^{K_{1B}}(0)$, where the contributions are included up to the first order in $m_{K_1}/m_b$. We resolve the sign ambiguity of the $K_1(1270)$--$K_1(1400)$ mixing angle $\\theta_{K_1}$ by defining the signs of decay constants, $f_{K_{1A}}$ and $f_{K_{1B}}^\\perp$. 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