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Category: Trigonometry

Question-170968

Question Number 170968 by 281981 last updated on 05/Jun/22 Commented by mr W last updated on 05/Jun/22 $$\mathrm{tan}^{\mathrm{2}} \:\left({x}+{y}\right)+\mathrm{cos}^{\mathrm{2}} \:\left({x}+{y}\right)−\mathrm{1}+\left({y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{tan}^{\mathrm{2}} \:\left({x}+{y}\right)−\mathrm{sin}^{\mathrm{2}} \:\left({x}+{y}\right)+\left({y}+\mathrm{1}\right)^{\mathrm{2}}…

Question-105366

Question Number 105366 by bemath last updated on 28/Jul/20 Answered by $@y@m last updated on 28/Jul/20 $$\mathrm{15}.\:\frac{\mathrm{tan}\:{A}\left(\mathrm{sec}\:{A}+\mathrm{1}\right)+\mathrm{tan}\:{A}\left(\mathrm{sec}\:{A}−\mathrm{1}\right)}{\mathrm{sec}^{\mathrm{2}} \:{A}−\mathrm{1}} \\ $$$$=\frac{\mathrm{2tan}\:{A}\mathrm{sec}\:{A}}{\mathrm{tan}\:^{\mathrm{2}} {A}} \\ $$$$=\mathrm{2cosec}\:{A} \\ $$…

Question-170872

Question Number 170872 by mathlove last updated on 02/Jun/22 Answered by aleks041103 last updated on 04/Jun/22 $$\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{{n}}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\frac{{e}^{\frac{{k}\pi{i}}{{n}}} −{e}^{−\frac{{k}\pi{i}}{{n}}} }{\mathrm{2}{i}}= \\ $$$$=\frac{\left(\underset{{k}=\mathrm{1}}…

Given-sin-2a-sin-2b-cos-2a-cos-2b-2-3-find-the-value-of-cos-a-b-

Question Number 105290 by bemath last updated on 27/Jul/20 $$\mathcal{G}{iven}\:\frac{\mathrm{sin}\:\mathrm{2}{a}−\mathrm{sin}\:\mathrm{2}{b}}{\mathrm{cos}\:\mathrm{2}{a}+\mathrm{cos}\:\mathrm{2}{b}}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${find}\:{the}\:{value}\:{of}\:\mathrm{cos}\:\left({a}−{b}\right)\: \\ $$ Answered by bobhans last updated on 27/Jul/20 $$\frac{\mathrm{2cos}\:\left(\frac{\mathrm{2}{a}+\mathrm{2}{b}}{\mathrm{2}}\right)\mathrm{sin}\:\left(\frac{\mathrm{2}{a}−\mathrm{2}{b}}{\mathrm{2}}\right)}{\mathrm{2cos}\:\left(\frac{\mathrm{2}{a}+\mathrm{2}{b}}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\mathrm{2}{a}−\mathrm{2}{b}}{\mathrm{2}}\right)}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{tan}\:\left({a}−{b}\right)\:=\:\frac{\mathrm{2}}{\mathrm{3}}\:\rightarrow\begin{cases}{\mathrm{sin}\:\left({a}−{b}\right)\:=\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{13}}}}\\{\mathrm{cos}\:\left({a}−{b}\right)\:=\:\frac{\mathrm{3}}{\:\sqrt{\mathrm{13}}}}\end{cases} \\…

sin-2x-sin-2y-4-9-cos-x-y-1-sin-x-y-0-lt-x-lt-pi-2-0-lt-y-lt-pi-2-find-the-value-of-sin-x-y-a-2-3-b-1-3-c-1-9-d-2-9-e-2-3-

Question Number 105243 by bemath last updated on 27/Jul/20 $$\begin{cases}{\mathrm{sin}\:\mathrm{2}{x}\:+\:\mathrm{sin}\:\mathrm{2}{y}\:=\:\frac{\mathrm{4}}{\mathrm{9}}}\\{\mathrm{cos}\:\left({x}−{y}\right)\:=\:\mathrm{1}−\mathrm{sin}\:\left({x}+{y}\right)}\end{cases} \\ $$$$\mathrm{0}\:<\:{x}\:<\:\frac{\pi}{\mathrm{2}}\:;\:\mathrm{0}\:<\:{y}\:<\:\frac{\pi}{\mathrm{2}} \\ $$$${find}\:{the}\:{value}\:{of}\:\mathrm{sin}\:\left({x}+{y}\right) \\ $$$$\left({a}\right)\:−\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\left({b}\right)\:−\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\left({c}\right)\:\frac{\mathrm{1}}{\mathrm{9}} \\ $$$$\left({d}\right)\:\frac{\mathrm{2}}{\mathrm{9}}\:\:\:\:\left({e}\right)\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$ Answered by bramlex last updated…

Solve-sinx-cosx-sinx-cosx-3-1-3-1-

Question Number 170752 by balirampatel last updated on 30/May/22 $${Solve}:−\:\:\frac{{sin}\boldsymbol{{x}}\:+\:{cos}\boldsymbol{{x}}}{{sin}\boldsymbol{{x}}\:−\:{cos}\boldsymbol{{x}}}\:=\:\frac{\sqrt{\mathrm{3}\:}\:−\:\mathrm{1}}{\:\sqrt{\mathrm{3}\:}\:+\:\mathrm{1}}\: \\ $$ Answered by Rasheed.Sindhi last updated on 30/May/22 $$\:\:\frac{{sin}\boldsymbol{{x}}\:+\:{cos}\boldsymbol{{x}}}{{sin}\boldsymbol{{x}}\:−\:{cos}\boldsymbol{{x}}}\:=\:\frac{\sqrt{\mathrm{3}\:}\:−\:\mathrm{1}}{\:\sqrt{\mathrm{3}\:}\:+\:\mathrm{1}}\: \\ $$$$\begin{array}{|c|}{\frac{{a}}{{b}}=\frac{{c}}{{d}}\Rightarrow\frac{{a}+{b}}{{a}−{b}}=\frac{{c}+{d}}{{c}−{d}}}\\\hline\end{array} \\ $$$$\:\:\frac{\left({sin}\boldsymbol{{x}}\:+\:{cos}\boldsymbol{{x}}\right)+\left({sin}\boldsymbol{{x}}\:−\:{cos}\boldsymbol{{x}}\right)}{\left({sin}\boldsymbol{{x}}\:+\:{cos}\boldsymbol{{x}}\right)−\left({sin}\boldsymbol{{x}}\:−\:{cos}\boldsymbol{{x}}\right)} \\…

2-3-sin-2-x-3pi-2-sin-2x-0-0-x-2pi-

Question Number 105112 by bemath last updated on 26/Jul/20 $$\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{sin}\:^{\mathrm{2}} \left({x}+\frac{\mathrm{3}\pi}{\mathrm{2}}\right)\:+\:\mathrm{sin}\:\mathrm{2}{x}\:=\:\mathrm{0}\: \\ $$$$\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi\: \\ $$ Answered by john santu last updated on 28/Jul/20 $$\Leftrightarrow\:\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{cos}\:^{\mathrm{2}} \left({x}\right)+\:\mathrm{sin}\:\left(\mathrm{2}{x}\right)\:=\:\mathrm{0}…