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Category: Trigonometry

If-3sin-x-4cos-x-5-then-sin-x-

Question Number 100131 by bemath last updated on 25/Jun/20 $$\mathrm{If}\:\mathrm{3sin}\:\mathrm{x}+\mathrm{4cos}\:\mathrm{x}=\mathrm{5}\:\mathrm{then}\:\mathrm{sin}\:\mathrm{x}=? \\ $$ Commented by bobhans last updated on 25/Jun/20 $$\mathrm{let}\:\mathrm{tan}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\:=\:\mathrm{t}\:,\:\Rightarrow\mathrm{sin}\:\mathrm{x}\:=\:\frac{\mathrm{2t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }\:,\:\mathrm{cos}\:\mathrm{x}=\frac{\mathrm{1}−\mathrm{t}^{\mathrm{2}} }{\mathrm{1}+\mathrm{t}^{\mathrm{2}} } \\ $$$$\Leftrightarrow\frac{\mathrm{6t}+\mathrm{4}−\mathrm{4t}^{\mathrm{2}}…

how-many-integer-number-satisfy-the-equation-sin-x-x-2-x-2-4x-5-0-

Question Number 100130 by bemath last updated on 25/Jun/20 $$\mathrm{how}\:\mathrm{many}\:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\frac{\mathrm{sin}\:\mathrm{x}−\mid\mathrm{x}+\mathrm{2}\mid}{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}−\mathrm{5}}\:\geqslant\:\mathrm{0}\: \\ $$ Answered by mr W last updated on 25/Jun/20…

37tanx-11tan3x-solve-it-

Question Number 165637 by daus last updated on 05/Feb/22 $$\mathrm{37}{tanx}\:=\mathrm{11}{tan}\mathrm{3}{x} \\ $$$${solve}\:{it} \\ $$ Answered by mr W last updated on 06/Feb/22 $$\mathrm{37}\:\mathrm{tan}\:{x}=\mathrm{11}×\frac{\mathrm{3}\:\mathrm{tan}\:{x}−\mathrm{tan}^{\mathrm{3}} \:{x}}{\mathrm{1}−\mathrm{3}\:\mathrm{tan}^{\mathrm{2}} \:{x}}…

Prove-that-sec-2-7-sec-4-7-sec-8-7-4-

Question Number 165558 by som(math1967) last updated on 03/Feb/22 $${Prove}\:{that} \\ $$$$\:\:\boldsymbol{{sec}}\frac{\mathrm{2}\boldsymbol{\pi}}{\mathrm{7}}\:+\boldsymbol{{sec}}\frac{\mathrm{4}\boldsymbol{\pi}}{\mathrm{7}}+\boldsymbol{{sec}}\frac{\mathrm{8}\boldsymbol{\pi}}{\mathrm{7}}\:=−\mathrm{4} \\ $$ Answered by Rohit143Jo last updated on 04/Feb/22 $${Ans}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{Let},\:{x}=\frac{\mathrm{2}\pi}{\mathrm{7}}\:\:\:\:\Leftrightarrow\:\:\mathrm{7}{x}=\mathrm{2}\pi \\…

Question-165428

Question Number 165428 by mathlove last updated on 01/Feb/22 Answered by som(math1967) last updated on 01/Feb/22 $$\:\frac{{sin}\mathrm{6}{sin}\mathrm{42}{sin}\mathrm{66}{sin}\mathrm{78}}{{cos}\mathrm{6}{cos}\mathrm{42}{cos}\mathrm{66}{cos}\mathrm{78}}\:\:\bigstar \\ $$$$=\frac{\left(\mathrm{2}{sin}\mathrm{6}{sin}\mathrm{66}\right)\left(\mathrm{2}{sin}\mathrm{42}{sin}\mathrm{78}\right)}{\left(\mathrm{2}{cos}\mathrm{6}{cos}\mathrm{66}\right)\left(\mathrm{2}{cos}\mathrm{42}{cos}\mathrm{78}\right)} \\ $$$$=\frac{\left\{{cos}\left(\mathrm{66}−\mathrm{6}\right)−{cos}\left(\mathrm{66}+\mathrm{6}\right)\right\}\left\{{cos}\mathrm{36}−{cos}\mathrm{120}\right\}}{\left({cos}\mathrm{72}+{cos}\mathrm{60}\right)\left({cos}\mathrm{120}+{cos}\mathrm{36}\right)} \\ $$$$=\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\right)}\:\:\:\bigstar\bigstar \\ $$$$=\frac{\left(\mathrm{3}−\sqrt{\mathrm{5}}\right)\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)}{\left(\sqrt{\mathrm{5}}+\mathrm{1}\right)\left(\sqrt{\mathrm{5}}−\mathrm{1}\right)}=\frac{\mathrm{9}−\mathrm{5}}{\mathrm{5}−\mathrm{1}}=\mathrm{1}…