If \(5\\sin y + \\cos y = \\sqrt{13}\\,\\sin y\), find the value of tan y.
\(\\dfrac{-5 - \\sqrt{13}}{12}\)
Rearrange: \(\\cos y = (\\sqrt{13} - 5)\\sin y\).
Divide both sides by cos y: \(1 = (\\sqrt{13} - 5)\\tan y\), so \(\\tan y = \\dfrac{1}{\\sqrt{13} - 5}\).
Rationalise: \(\\tan y = \\dfrac{1}{\\sqrt{13} - 5} \\times \\dfrac{\\sqrt{13} + 5}{\\sqrt{13} + 5} = \\dfrac{\\sqrt{13} + 5}{13 - 25} = \\dfrac{\\sqrt{13} + 5}{-12} = \\dfrac{-5 - \\sqrt{13}}{12}\).
Hence, \(\\tan y = \\dfrac{-5 - \\sqrt{13}}{12}\).
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