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Let-be-an-acute-angle-such-that-the-equation-x-2-4xcos-cot-0-involving-variable-x-has-multiple-roots-Then-the-measure-of-in-radians-is-




Question Number 144800 by imjagoll last updated on 29/Jun/21
Let β be an acute angle such  that the equation x^2 +4xcos β+cot β=0  involving variable x has multiple  roots. Then the measure of β in  radians is __
$$\mathrm{Let}\:\beta\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{angle}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{4xcos}\:\beta+\mathrm{cot}\:\beta=\mathrm{0} \\ $$$$\mathrm{involving}\:\mathrm{variable}\:\mathrm{x}\:\mathrm{has}\:\mathrm{multiple} \\ $$$$\mathrm{roots}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{measure}\:\mathrm{of}\:\beta\:\mathrm{in} \\ $$$$\mathrm{radians}\:\mathrm{is}\:\_\_ \\ $$
Answered by liberty last updated on 29/Jun/21
 since the equation x^2 +4x cos β+cot β=0  has multiple roots , we have  △=0→16cos^2 β−4cot β=0  →4cot β(2sin 2β−1)=0  when 0<β<(π/2) we get    sin 2β=(1/2) so  { ((β=(π/(12)) or)),((β=((5π)/(12)))) :}.
$$\:\mathrm{since}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{4x}\:\mathrm{cos}\:\beta+\mathrm{cot}\:\beta=\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{multiple}\:\mathrm{roots}\:,\:\mathrm{we}\:\mathrm{have} \\ $$$$\bigtriangleup=\mathrm{0}\rightarrow\mathrm{16cos}\:^{\mathrm{2}} \beta−\mathrm{4cot}\:\beta=\mathrm{0} \\ $$$$\rightarrow\mathrm{4cot}\:\beta\left(\mathrm{2sin}\:\mathrm{2}\beta−\mathrm{1}\right)=\mathrm{0} \\ $$$$\mathrm{when}\:\mathrm{0}<\beta<\frac{\pi}{\mathrm{2}}\:\mathrm{we}\:\mathrm{get}\: \\ $$$$\:\mathrm{sin}\:\mathrm{2}\beta=\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{so}\:\begin{cases}{\beta=\frac{\pi}{\mathrm{12}}\:\mathrm{or}}\\{\beta=\frac{\mathrm{5}\pi}{\mathrm{12}}}\end{cases}. \\ $$$$ \\ $$

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