Question Number 193669 by Rupesh123 last updated on 17/Jun/23 Answered by AST last updated on 18/Jun/23 $${cos}\left({x}+{y}\right)+{isin}\left({x}+{y}\right)={e}^{{i}\left({x}+{y}\right)} ={e}^{{ix}} {e}^{{iy}} \\ $$$$=\left({cos}\left({x}\right)+{isin}\left({x}\right)\right)\left({cos}\left({y}\right)+{isin}\left({y}\right)\right) \\ $$$$={cos}\left({x}\right){cos}\left({y}\right)−{sin}\left({x}\right){cos}\left({y}\right)+{icos}\left({x}\right){sin}\left({y}\right)+{isin}\left({x}\right){cos}\left({y}\right) \\ $$$${Comparing}\:{real}\:{parts}…
Question Number 193668 by SAMIRA last updated on 17/Jun/23 $$\frac{\mathrm{sin}\left(\mathrm{3x}\right)}{\mathrm{1}−\mathrm{x}}−\frac{\mathrm{cos}\left(\mathrm{3x}\right)}{\mathrm{cos}\left(\mathrm{x}\right)}\:=\:−\mathrm{2}\:\:????\:\: \\ $$ Answered by Frix last updated on 18/Jun/23 $$\mathrm{Transform}\:\mathrm{to} \\ $$$${x}=−\frac{\mathrm{4sin}^{\mathrm{3}} \:{x}\:−\mathrm{4sin}^{\mathrm{2}} \:{x}\:−\mathrm{3sin}\:{x}\:−\mathrm{1}}{\mathrm{1}+\mathrm{4sin}^{\mathrm{2}} \:{x}}={f}\left({x}\right)…
Question Number 193533 by horsebrand11 last updated on 16/Jun/23 $$\:\:\:\mathrm{If}\:\mathrm{cot}\:\mathrm{x}−\mathrm{tan}\:\mathrm{x}=\mathrm{4}\:\mathrm{then} \\ $$$$\:\:\:\:\mathrm{cot}\:^{\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{2x}}\:−\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}\:=? \\ $$ Answered by MM42 last updated on 16/Jun/23 $$\mathrm{1}−{tan}^{\mathrm{2}} {x}=\mathrm{4}{tanx}\Rightarrow{tan}^{\mathrm{2}}…
Question Number 193502 by horsebrand11 last updated on 15/Jun/23 $$\:\cancel{\underline{{o}}}\mathrm{olve}\: \\ $$$$\:\:\mathrm{cos}\:\mathrm{2x}\:.\mathrm{tan}\:\left(\frac{\mathrm{7}\pi}{\mathrm{19}}\right)=\mathrm{tan}\:\left(\frac{\mathrm{17}\pi}{\mathrm{23}}\right)+\mathrm{tan}\:\left(\frac{\mathrm{6}\pi}{\mathrm{23}}\right)+\mathrm{tan}\:\left(\frac{\mathrm{12}\pi}{\mathrm{19}}\right) \\ $$ Answered by cortano12 last updated on 15/Jun/23 $$\:\:\mathrm{If}\:\alpha+\beta=\pi\:\mathrm{then}\:\begin{cases}{\mathrm{tan}\:\alpha+\mathrm{tan}\:\beta=\mathrm{0}}\\{\mathrm{tan}\:\alpha\:\mathrm{cot}\:\beta=−\mathrm{1}}\end{cases} \\ $$$$\:\mathrm{so}\:\mathrm{cos}\:\:\mathrm{2x}\:\mathrm{tan}\:\left(\frac{\mathrm{7}\pi}{\mathrm{19}}\right)=\mathrm{0}+\mathrm{tan}\:\left(\frac{\mathrm{12}\pi}{\mathrm{19}}\right) \\…
Question Number 193464 by Mingma last updated on 14/Jun/23 Commented by Frix last updated on 14/Jun/23 $$\mathrm{For}\:\mathrm{any}\:\mathrm{triangle} \\ $$$${r}_{{n}} =\frac{{c}\delta}{\mathrm{2}\left(\left({a}+{b}+{c}\right){c}+\left({n}−\mathrm{1}\right)\delta\right)} \\ $$$$\:\:\:\:\:\left[\delta=\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}\right. \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{rectangular}\:\mathrm{triangle}\:\mathrm{with}\:{c}=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}}…
Question Number 193461 by SAMIRA last updated on 14/Jun/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}−\mathrm{1}}{\mathrm{sin}\:\mathrm{2x}}\right)\:=\:?? \\ $$ Answered by aba last updated on 14/Jun/23 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}+\mathrm{sinx}−\mathrm{1}}{\mathrm{sin}\left(\mathrm{2x}\right)\left(\sqrt{\mathrm{1}+\mathrm{sinx}}+\mathrm{1}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{2cosx}\left(\sqrt{\mathrm{1}+\mathrm{sinx}}+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{4}}\:\checkmark \\ $$…
Question Number 193458 by SAMIRA last updated on 14/Jun/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}\:\mathrm{sin}\left(\mathrm{x}\right)}\right)\:=\:\:\:??? \\ $$ Commented by aba last updated on 14/Jun/23 $$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }×\frac{\mathrm{x}}{\mathrm{sin}\left(\mathrm{x}\right)}=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{1}=\mathrm{0}.\mathrm{5} \\…
Question Number 193436 by beto last updated on 14/Jun/23 $$ \\ $$$$\frac{{sin}\left({x}+\mathrm{18}^{{o}} \right)}{{sin}\left(\mathrm{48}^{{o}} \right)}=\frac{{sin}\left({x}\right)}{{sin}\left(\mathrm{18}^{{o}} \right)} \\ $$$$ \\ $$ Answered by a.lgnaoui last updated on…
Question Number 193423 by mustafazaheen last updated on 13/Jun/23 $$\mathrm{when}\:\:\:\mathrm{tan}\frac{\theta}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{a}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{cos}\theta=?\:\mathrm{from}\:\mathrm{the}\:\mathrm{a} \\ $$ Answered by AST last updated on 13/Jun/23 $${tan}\left(\frac{\theta}{\mathrm{2}}+\frac{\theta}{\mathrm{2}}\right)=\frac{\mathrm{2}{tan}\left(\frac{\theta}{\mathrm{2}}\right)}{\mathrm{1}−{tan}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}\Rightarrow{tan}\left(\theta\right)=\frac{\mathrm{2}{a}}{{a}^{\mathrm{2}} −\mathrm{1}} \\…
Question Number 193399 by Mingma last updated on 12/Jun/23 Answered by MathematicalUser2357 last updated on 10/Sep/23 $${A}=\mathrm{cosec}\:\mathrm{39}°\approx\mathrm{1}.\mathrm{589016} \\ $$$${B}=\phi=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\approx\mathrm{1}.\mathrm{618034} \\ $$$$\mathrm{So},\:{A}\rangle{B} \\ $$ Terms of…