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Find-5-sin-pi-6-1-tg-75-tg-75-




Question Number 207157 by hardmath last updated on 07/May/24
Find:   ((5 sin (π/6))/((1/(tg 75°))  −  tg 75°))  =  ?
$$\mathrm{Find}:\:\:\:\frac{\mathrm{5}\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}}}{\frac{\mathrm{1}}{\mathrm{tg}\:\mathrm{75}°}\:\:−\:\:\mathrm{tg}\:\mathrm{75}°}\:\:=\:\:? \\ $$
Answered by A5T last updated on 07/May/24
(1/(tan75))−tan75=((cos75)/(sin75))−((sin75)/(cos75))=((cos^2 75−sin^2 75)/(sin75cos75))  =((2cos150)/(sin150))=((2(((−(√3))/2)))/(1/2))=−2(√3)  ⇒?=((5sin((π/6)))/(−2(√3)))=((5/2)/(−2(√3)))=((−5(√3))/(12))
$$\frac{\mathrm{1}}{{tan}\mathrm{75}}−{tan}\mathrm{75}=\frac{{cos}\mathrm{75}}{{sin}\mathrm{75}}−\frac{{sin}\mathrm{75}}{{cos}\mathrm{75}}=\frac{{cos}^{\mathrm{2}} \mathrm{75}−{sin}^{\mathrm{2}} \mathrm{75}}{{sin}\mathrm{75}{cos}\mathrm{75}} \\ $$$$=\frac{\mathrm{2}{cos}\mathrm{150}}{{sin}\mathrm{150}}=\frac{\mathrm{2}\left(\frac{−\sqrt{\mathrm{3}}}{\mathrm{2}}\right)}{\frac{\mathrm{1}}{\mathrm{2}}}=−\mathrm{2}\sqrt{\mathrm{3}} \\ $$$$\Rightarrow?=\frac{\mathrm{5}{sin}\left(\frac{\pi}{\mathrm{6}}\right)}{−\mathrm{2}\sqrt{\mathrm{3}}}=\frac{\frac{\mathrm{5}}{\mathrm{2}}}{−\mathrm{2}\sqrt{\mathrm{3}}}=\frac{−\mathrm{5}\sqrt{\mathrm{3}}}{\mathrm{12}} \\ $$

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