Question Number 18546 by Tinkutara last updated on 24/Jul/17 $$\mathrm{In}\:\Delta{ABC},\:\mathrm{tan}\frac{{A}}{\mathrm{2}}\:+\:\mathrm{tan}\frac{{B}}{\mathrm{2}}\:+\:\mathrm{tan}\frac{{C}}{\mathrm{2}}\:=\:\sqrt{\mathrm{3}}, \\ $$$$\mathrm{then}\:\Delta\:\mathrm{must}\:\mathrm{be} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Equilateral} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Isosceles} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Acute}\:\mathrm{angled} \\ $$ Commented by b.e.h.i.8.3.417@gmail.com last updated…
Question Number 18545 by Tinkutara last updated on 24/Jul/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{sin3}{x}\:+\:\mathrm{cos2}{x}\:=\:\mathrm{0}\:\mathrm{in}\:\left[\mathrm{0},\:\frac{\mathrm{3}\pi}{\mathrm{2}}\right]\:\mathrm{is} \\ $$ Answered by 433 last updated on 24/Jul/17 $$ \\ $$$$ \\…
Question Number 18523 by Tinkutara last updated on 23/Jul/17 $$\mathrm{Match}\:\mathrm{the}\:\mathrm{following} \\ $$$$\boldsymbol{\mathrm{Column}}-\boldsymbol{\mathrm{I}}\:\left(\boldsymbol{\mathrm{Trigonometric}}\:\boldsymbol{\mathrm{equation}}\right) \\ $$$$\left(\mathrm{A}\right)\:\mathrm{sin}\:\mathrm{9}\theta\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}\:−\:\theta\right) \\ $$$$\left(\mathrm{B}\right)\:\mathrm{sin}\:\mathrm{5}\theta\:=\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}\:+\:\mathrm{2}\theta\right) \\ $$$$\left(\mathrm{C}\right)\:\mathrm{cos}\:\mathrm{11}\theta\:=\:\mathrm{cos}\:\mathrm{3}\theta \\ $$$$\left(\mathrm{D}\right)\:\mathrm{3}\:\mathrm{tan}\:\left(\theta\:−\:\mathrm{15}°\right)\:=\:\mathrm{tan}\:\left(\theta\:+\:\mathrm{15}°\right) \\ $$$$\boldsymbol{\mathrm{Column}}-\boldsymbol{\mathrm{II}}\:\left(\boldsymbol{\mathrm{Family}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{solutions}}\right) \\ $$$$\left(\mathrm{p}\right)\:\left(\mathrm{2}{n}\:+\:\mathrm{1}\right)\frac{\pi}{\mathrm{10}},\:{n}\:\in\:{Z} \\…
Question Number 18524 by Tinkutara last updated on 23/Jul/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{sin}^{\mathrm{3}} {x}\:−\:\mathrm{3sin}{x}\mathrm{cos}^{\mathrm{2}} {x}\:+\:\mathrm{2cos}^{\mathrm{3}} {x}\:=\:\mathrm{0}\:\mathrm{in} \\ $$$$\left[−\frac{\pi}{\mathrm{4}},\:\frac{\pi}{\mathrm{4}}\right]\:\mathrm{is} \\ $$ Answered by Tinkutara last updated on…
Question Number 18501 by Tinkutara last updated on 22/Jul/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{\Delta}{\mathrm{2}{R}^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{{r}}{\mathrm{2}{R}}\left(\mathrm{sin}{A}\:+\:\mathrm{sin}{B}\:+\:\mathrm{sin}{C}\right) \\ $$$$\left(\mathrm{3}\right)\:{a}\mathrm{cos}{A}\:+\:{b}\mathrm{cos}{B}\:+\:{c}\mathrm{cos}{C}\:=\:\frac{{abc}}{\mathrm{2}{R}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\mathrm{sin}{A}.\mathrm{sin}{B}.\mathrm{sin}{C}\:=\:\frac{{R}}{\mathrm{2}{r}}\left(\mathrm{sin}{A}\:+\:\mathrm{sin}{B}\:+\:\mathrm{sin}{C}\right) \\ $$ Commented by b.e.h.i.8.3.417@gmail.com…
Question Number 84019 by mathocean1 last updated on 08/Mar/20 $${Show}\:{that}: \\ $$$$\mathrm{1}\bullet\:\:\:{tan}\mathrm{3}{x}=\frac{\mathrm{3}−{tan}^{\mathrm{2}} {x}}{\mathrm{1}−\mathrm{3}{tan}^{\mathrm{2}} {x}} \\ $$$${using}\:{cos}\mathrm{3}{x}=\mathrm{4}{cos}^{\mathrm{4}} {x}−\mathrm{3}{cosx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{sin}\mathrm{3}{x}=−\mathrm{4}{sin}^{\mathrm{3}} {x}+\mathrm{3}{sinx} \\ $$$${Thanks}… \\ $$ Commented…
Question Number 18474 by Tinkutara last updated on 22/Jul/17 $$\boldsymbol{\mathrm{Assertion}}-\boldsymbol{\mathrm{Reason}}\:\boldsymbol{\mathrm{Type}}\:\boldsymbol{\mathrm{Question}} \\ $$$$\mathrm{STATEMENT}-\mathrm{1}\::\:{f}\left({x}\right)\:=\:\mathrm{log}_{\mathrm{cos}{x}} \mathrm{sin}{x}\:\mathrm{is} \\ $$$$\mathrm{well}\:\mathrm{defined}\:\mathrm{in}\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right). \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mathrm{sin}{x}\:\mathrm{and}\:\mathrm{cos}{x}\:\mathrm{are} \\ $$$$\mathrm{positive}\:\mathrm{in}\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right). \\ $$ Answered by…
Question Number 18472 by Tinkutara last updated on 22/Jul/17 $$\mathrm{The}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{2}^{\mathrm{sin}\:{x}} \:+\:\mathrm{2}^{\mathrm{cos}\:{x}} \\ $$$$=\:\mathrm{2}^{\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}} \:\mathrm{is} \\ $$ Answered by sushmitak last updated on 22/Jul/17 $$\mathrm{sin}\:{x}\mathrm{cos}\:{x}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}=\frac{\mathrm{1}}{\mathrm{2}} \\…
Question Number 18457 by Tinkutara last updated on 21/Jul/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{sin}\:\theta\:+\:\mathrm{cos}\:\theta\:=\:\mathrm{1}\:+\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{interval}\:\left[\mathrm{0},\:\mathrm{4}\pi\right]\:\mathrm{is} \\ $$ Answered by mrW1 last updated on 21/Jul/17 $$\mathrm{sin}\:\theta\:+\:\mathrm{cos}\:\theta\:=\:\mathrm{1}\:+\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta \\…
Question Number 18455 by Tinkutara last updated on 21/Jul/17 $$\mathrm{The}\:\mathrm{equation}\:\mathrm{cosec}\:\frac{{x}}{\mathrm{2}}\:+\:\mathrm{cosec}\:\frac{{y}}{\mathrm{2}}\:+ \\ $$$$\mathrm{cosec}\:\frac{{z}}{\mathrm{2}}\:=\:\mathrm{6},\:\mathrm{where}\:\mathrm{0}\:<\:{x},\:{y},\:{z}\:<\:\frac{\pi}{\mathrm{2}}\:\mathrm{and} \\ $$$${x}\:+\:{y}\:+\:{z}\:=\:\pi,\:\mathrm{have} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Three}\:\mathrm{ordered}\:\mathrm{triplet}\:\left({x},\:{y},\:{z}\right) \\ $$$$\mathrm{solutions} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Two}\:\mathrm{ordered}\:\mathrm{triplet}\:\left({x},\:{y},\:{z}\right) \\ $$$$\mathrm{solutions} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Just}\:\mathrm{one}\:\mathrm{ordered}\:\mathrm{triplet}\:\left({x},\:{y},\:{z}\right) \\…