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Category: Trigonometry

tan-2x-tan-3x-tan-5x-1-

Question Number 156887 by cortano last updated on 16/Oct/21 $$\:\mathrm{tan}\:\mathrm{2}{x}\:\mathrm{tan}\:\mathrm{3}{x}\:\mathrm{tan}\:\mathrm{5}{x}\:=\mathrm{1} \\ $$ Answered by peter frank last updated on 18/Oct/21 $$\mathrm{tan}\:\left(\mathrm{x}+\mathrm{y}\right)=\frac{\mathrm{tan}\:\mathrm{x}+\mathrm{tan}\:\mathrm{y}}{\mathrm{1}−\mathrm{tan}\:\mathrm{xtan}\:\mathrm{y}} \\ $$$$\mathrm{tan}\:\mathrm{5x}=\mathrm{tan}\:\left(\mathrm{2x}+\mathrm{3x}\right)=\frac{\mathrm{tan}\:\mathrm{2x}+\mathrm{tan}\:\mathrm{3x}}{\mathrm{1}−\mathrm{tan}\:\mathrm{2xtan}\:\mathrm{3y}} \\ $$$$\mathrm{tan}\:\mathrm{5x}−\mathrm{tan}\:\mathrm{5xtan}\:\mathrm{2xtan}\:\mathrm{3x}=\mathrm{tan}\:\mathrm{2x}+\mathrm{tan}\:\mathrm{3x}…

show-that-cos-50-sin-40-sin-42-cos-48-2tan-18-cot-72-0-

Question Number 25771 by keyurpatel last updated on 14/Dec/17 $${show}\:{that}\:\frac{\mathrm{cos}\:\mathrm{50}}{\mathrm{sin}\:\mathrm{40}}+\frac{\mathrm{sin}\:\mathrm{42}}{\mathrm{cos}\:\mathrm{48}}−\frac{\mathrm{2tan}\:\mathrm{18}}{\mathrm{cot}\:\mathrm{72}}=\mathrm{0} \\ $$ Answered by deepak123 last updated on 14/Dec/17 $$\mathrm{sin}\:\left(\mathrm{90}−\theta\right)=\mathrm{cos}\theta \\ $$$$\mathrm{cos}\:\left(\mathrm{90}−\theta\right)=\mathrm{sin}\:\theta \\ $$$$\mathrm{tan}\:\left(\mathrm{90}−\theta\right)=\mathrm{cot}\:\theta \\…

1-cos-2-2-2-cos-4-sin-4-

Question Number 25763 by amankumar last updated on 14/Dec/17 $$\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{2}\theta=\mathrm{2}\left(\mathrm{cos}\:^{\mathrm{4}} \theta+\mathrm{sin}\:^{\mathrm{4}} \theta\right) \\ $$ Answered by ajfour last updated on 15/Dec/17 $$\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{2}\theta=\left(\mathrm{sin}\:^{\mathrm{2}} \theta+\mathrm{cos}\:^{\mathrm{2}}…

f-x-arctg-1-x-2-x-1-and-f-1-f-2-f-21-find-tg-

Question Number 156807 by amin96 last updated on 15/Oct/21 $${f}\left({x}\right)={arctg}\frac{\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:\:{and}\:\alpha={f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+\ldots+{f}\left(\mathrm{21}\right) \\ $$$${find}\:\:{tg}\left(\alpha\right)=? \\ $$ Commented by amin96 last updated on 15/Oct/21 $$\left.{A}\left.\right)\left.\frac{\mathrm{6}}{\mathrm{11}}\left.\:\left.\:\:\:{B}\right)\frac{\mathrm{7}}{\mathrm{11}}\:\:\:{C}\right)\frac{\mathrm{11}}{\mathrm{21}}\:\:\:\:{D}\right)\frac{\mathrm{1}}{\mathrm{21}}\:\:\:\:{E}\right)\frac{\mathrm{21}}{\mathrm{23}} \\ $$…

if-cosh-u-iv-x-iy-show-that-x-2-cosh-2-u-y-2-sinh-2-u-1-and-x-2-cosh-2-v-y-2-sinh-2-v-1-where-sinhx-and-coshx-hhyperbolcfunction-

Question Number 25611 by nnnavendu last updated on 12/Dec/17 $${if}\:{cosh}\left({u}+{iv}\right)={x}+{iy}\:,\:{show}\:{that} \\ $$$$\frac{{x}^{\mathrm{2}} }{{cosh}^{\mathrm{2}} {u}}+\frac{{y}^{\mathrm{2}} }{{sinh}^{\mathrm{2}} {u}}=\mathrm{1}\:\:{and}\:\:\frac{{x}^{\mathrm{2}} }{{cosh}^{\mathrm{2}} {v}}−\frac{{y}^{\mathrm{2}} }{{sinh}^{\mathrm{2}} {v}}=\mathrm{1} \\ $$$$ \\ $$$${where}\:\:\:\:{sinhx}\:{and}\:{coshx}\:\:\:\:{hhyperbolcfunction} \\…

2cos-pi-9-1-3-2cos-2pi-9-1-3-2cos-4pi-9-1-3-

Question Number 90876 by john santu last updated on 26/Apr/20 $$\sqrt[{\mathrm{3}\:\:}]{\mathrm{2cos}\:\left(\frac{\pi}{\mathrm{9}}\right)}−\sqrt[{\mathrm{3}\:\:}]{\mathrm{2cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)}−\sqrt[{\mathrm{3}\:\:}]{\mathrm{2cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{9}}\right)}\:=\:? \\ $$ Commented by john santu last updated on 26/Apr/20 $${Ramanujan}\:{theorem} \\ $$$${let}\:\alpha,\:\beta\:,\gamma\:{be}\:{a}\:{roots}\: \\…

Question-25343

Question Number 25343 by Tinkutara last updated on 08/Dec/17 Commented by sushmitak last updated on 08/Dec/17 $${As}\:{pointed}\:{out}\:{below}.\:{This} \\ $$$${is}\:{incorrect}\:{logic}. \\ $$$${The}\:{correct}\:{logic}\:{for}\:{interchangable} \\ $$$${variables}\:{in}\:{an}\:{equation}\:{is}\:{that} \\ $$$${if}\:{p}\:{is}\:{a}\:{solutoon}\:{for}\:{b}\:{then}\:{it}…

cos-4-pi-9-cos-4-2pi-9-cos-4-3pi-9-cos-4-4pi-9-

Question Number 156321 by cortano last updated on 10/Oct/21 $$\:\:\mathrm{cos}\:^{\mathrm{4}} \left(\frac{\pi}{\mathrm{9}}\right)+\mathrm{cos}\:^{\mathrm{4}} \left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)+\mathrm{cos}\:^{\mathrm{4}} \left(\frac{\mathrm{3}\pi}{\mathrm{9}}\right)+\mathrm{cos}\:^{\mathrm{4}} \left(\frac{\mathrm{4}\pi}{\mathrm{9}}\right)=? \\ $$ Commented by john_santu last updated on 10/Oct/21 $${ans}\::\:\frac{\mathrm{19}}{\mathrm{16}} \\…