Question Number 209838 by malwan last updated on 23/Jul/24
$${If}\:{sin}\:{x}\:+\:{cos}\:{x}\:+\:{tan}\:{x}\:+\:{cot}\:{x}\:+ \\ $$$${sec}\:{x}\:+\:{csc}\:{x}\:=\:\mathrm{7}\:,\:{find}\:{sin}\mathrm{2}{x}? \\ $$
Answered by mahdipoor last updated on 23/Jul/24
![=((s^2 c+c^2 s+s^2 +c^2 +c+s)/(sc))=((1+(sc+1)(c+s))/(sc))= ((1+(((sin(2x))/2)+1)(cosx+sinx))/((sin(2x))/2))=7 (eq i) get sin(2x)=2u ⇒sinx+cosx=(√(sin^2 x+cos^2 x+2sin(x)cos(x)))= (√(1+sin(2x)))=(√(1+2u)) eq i⇒((1+(u+1)(√(1+2u)))/u)=7⇒^(u≠0) (√(1+2u))=((7u−1)/(u+1))⇒ 1+2u=((49u^2 +1−14u)/(u^2 +1+2u))⇒2u^3 −44u^2 +18u=0 u=0(reject),11±4(√7) ⇒ −1≤sin(2x)=2u≤1 ⇒sin(2x)=2u=22−8(√7)=0.8340 (its mean [0∼360] : x=28,26 , 61.74 deg)](https://www.tinkutara.com/question/Q209843.png)
$$=\frac{{s}^{\mathrm{2}} {c}+{c}^{\mathrm{2}} {s}+{s}^{\mathrm{2}} +{c}^{\mathrm{2}} +{c}+{s}}{{sc}}=\frac{\mathrm{1}+\left({sc}+\mathrm{1}\right)\left({c}+{s}\right)}{{sc}}= \\ $$$$\frac{\mathrm{1}+\left(\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}}+\mathrm{1}\right)\left({cosx}+{sinx}\right)}{\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}}}=\mathrm{7}\:\:\:\:\left({eq}\:{i}\right) \\ $$$${get}\:{sin}\left(\mathrm{2}{x}\right)=\mathrm{2}{u} \\ $$$$\Rightarrow{sinx}+{cosx}=\sqrt{{sin}^{\mathrm{2}} {x}+{cos}^{\mathrm{2}} {x}+\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right)}= \\ $$$$\sqrt{\mathrm{1}+{sin}\left(\mathrm{2}{x}\right)}=\sqrt{\mathrm{1}+\mathrm{2}{u}} \\ $$$${eq}\:{i}\Rightarrow\frac{\mathrm{1}+\left({u}+\mathrm{1}\right)\sqrt{\mathrm{1}+\mathrm{2}{u}}}{{u}}=\mathrm{7}\overset{{u}\neq\mathrm{0}} {\Rightarrow}\sqrt{\mathrm{1}+\mathrm{2}{u}}=\frac{\mathrm{7}{u}−\mathrm{1}}{{u}+\mathrm{1}}\Rightarrow \\ $$$$\mathrm{1}+\mathrm{2}{u}=\frac{\mathrm{49}{u}^{\mathrm{2}} +\mathrm{1}−\mathrm{14}{u}}{{u}^{\mathrm{2}} +\mathrm{1}+\mathrm{2}{u}}\Rightarrow\mathrm{2}{u}^{\mathrm{3}} −\mathrm{44}{u}^{\mathrm{2}} +\mathrm{18}{u}=\mathrm{0} \\ $$$${u}=\mathrm{0}\left({reject}\right),\mathrm{11}\pm\mathrm{4}\sqrt{\mathrm{7}}\:\Rightarrow\:−\mathrm{1}\leqslant{sin}\left(\mathrm{2}{x}\right)=\mathrm{2}{u}\leqslant\mathrm{1}\:\: \\ $$$$\Rightarrow{sin}\left(\mathrm{2}{x}\right)=\mathrm{2}{u}=\mathrm{22}−\mathrm{8}\sqrt{\mathrm{7}}=\mathrm{0}.\mathrm{8340} \\ $$$$\left({its}\:{mean}\:\left[\mathrm{0}\sim\mathrm{360}\right]\::\:{x}=\mathrm{28},\mathrm{26}\:\:,\:\:\mathrm{61}.\mathrm{74}\:\:{deg}\right) \\ $$
Commented by malwan last updated on 23/Jul/24
