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Question Number 216369 by AROUNAMoussa last updated on 05/Feb/25
Answered by AROUNAMoussa last updated on 05/Feb/25
Determine: mesIB^� N puis mesJM^� P
$${Determine}:\:{mesI}\hat {{B}N}\:{puis}\:{mesJ}\hat {{M}P} \\ $$
Commented by a.lgnaoui last updated on 06/Feb/25
$$ \\ $$
Answered by a.lgnaoui last updated on 06/Feb/25
mes(JNP)=mes(JIP)=57
$$\mathrm{mes}\left(\mathrm{JNP}\right)=\mathrm{mes}\left(\mathrm{JIP}\right)=\mathrm{57} \\ $$$$ \\ $$
Commented by AROUNAMoussa last updated on 06/Feb/25
et mes(IBN)=? mes(JMP)=?
$${et}\:{mes}\left({IBN}\right)=?\:\:\:{mes}\left({JMP}\right)=? \\ $$
Commented by a.lgnaoui last updated on 08/Feb/25
IN^2 =MI^2 +MN^2 −2MN.MIcos (57−x) MP=PIcos x+MIcos (57−x) (1) MJ=NJcos x+MNcos (57−x)(2) MI.MJ=MN.MP ((MP)/(MJ))=((MI)/(MN)) ((MP)/(MJ))=((PIcos x+MI(cos 57cos x+sin 57sin x))/(NJcos x+MN(cos 57cos x+sin 57sin x))) =(([PI+MI(cos 57+sin 57tan x))/(NJ+MN(cos 57+sin 57tan x)))=((MI)/(MN)) MN×PI+MN.MI(cos 57+sin 57tan x) =MI.NJ+MI.MN(cos 57+sin 57tan x) MN.PI=MI.NJ ?((MI)/(MN))=((PI)/(NJ))=((MP)/(MJ)) MI=((MN×PI)/(NJ)) ((sin x)/(MI))=sin x( ((sin (57/t−cos 57))/(PI))) (1/(MI))=((sin 57/t−cos 57))/(PI)) MI/PI=MN/NJ= MI^2 +PI^2 −2MI.PIcos 123=MP^2 MI^2 +MI^2 (sin 57/t−cos 57) −2.MI^2 .(sin 57/t−cos 57)cos 123=MP^2 =MI^2 [1+(sin57/t −cos 57)−2(sin 57/t−cos57 )cos 123 MI^2 [(1+(sin57/t −cos57 )(1−2cos 123)]=MP^2 (((MP)/(MI)))^2 =[1+(sin 57/t−cos 57)(1−cos 123)] ((sin 123)/(MP))=((sin x)/(MI)) ⇒ ((MP)/(MI))=((sin 123)/(sin x)) donc ((sin^2 123)/(sin^2 x))=(1+((sin 57)/t)−cos 57)(1−cos 123) (1/(sin^2 x))=1+(1/t^2 ) ((sin^2 123(1+t^2 ))/t^2 )=(1−cos 223)(((sin 57+t−tcos 57)/t^2 ))t (1−cos 223)(t^2 (1−cos 57)+sin 57t −sin^2 123×t^2 −sin^2 123=0 (1−cos 57)(1−cos 123)t^2 +sin 57(1−cos 223)t −sin^2 123t^2 −sin^2 123=0 l equation finale est: [(1−Cos 57)(1−cos 123)−sin^2 123)]t^2 +sin 57(1−cos 123)t−sin^2 123=0 (1−cos^2 57−sin^2 57)t^2 =0 (sin 57+sin 57cos 57)t=sin^2 57 soit t=((sin 57)/((1+cos 57))) x=28,5 ⇒ { ((∡IBN=57+x=85,5)),((∡JMP=57−x=28,5°)) :}
$$\mathrm{IN}^{\mathrm{2}} =\mathrm{MI}^{\mathrm{2}} +\mathrm{MN}^{\mathrm{2}} −\mathrm{2MN}.\mathrm{MIcos}\:\left(\mathrm{57}−\boldsymbol{\mathrm{x}}\right) \\ $$$$ \\ $$$$\boldsymbol{\mathrm{MP}}=\boldsymbol{\mathrm{PI}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{MI}}\mathrm{cos}\:\left(\mathrm{57}−\boldsymbol{\mathrm{x}}\right)\:\:\left(\mathrm{1}\right) \\ $$$$\boldsymbol{\mathrm{MJ}}=\boldsymbol{\mathrm{NJ}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{MN}}\mathrm{cos}\:\left(\mathrm{57}−\boldsymbol{\mathrm{x}}\right)\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\boldsymbol{\mathrm{MI}}.\boldsymbol{\mathrm{MJ}}=\boldsymbol{\mathrm{MN}}.\boldsymbol{\mathrm{MP}} \\ $$$$\frac{\boldsymbol{\mathrm{MP}}}{\boldsymbol{\mathrm{MJ}}}=\frac{\boldsymbol{\mathrm{MI}}}{\boldsymbol{\mathrm{MN}}} \\ $$$$\frac{\boldsymbol{\mathrm{MP}}}{\boldsymbol{\mathrm{MJ}}}=\frac{\boldsymbol{\mathrm{PI}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{MI}}\left(\mathrm{cos}\:\mathrm{57cos}\:\boldsymbol{\mathrm{x}}+\mathrm{sin}\:\mathrm{57sin}\:\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{NJ}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{MN}}\left(\mathrm{cos}\:\mathrm{57cos}\:\boldsymbol{\mathrm{x}}+\mathrm{sin}\:\mathrm{57sin}\:\boldsymbol{\mathrm{x}}\right)} \\ $$$$=\frac{\left[\boldsymbol{\mathrm{PI}}+\boldsymbol{\mathrm{MI}}\left(\mathrm{cos}\:\mathrm{57}+\mathrm{sin}\:\mathrm{57tan}\:\boldsymbol{\mathrm{x}}\right)\right.}{\boldsymbol{\mathrm{NJ}}+\boldsymbol{\mathrm{MN}}\left(\mathrm{cos}\:\mathrm{57}+\mathrm{sin}\:\mathrm{57tan}\:\boldsymbol{\mathrm{x}}\right)}=\frac{\boldsymbol{\mathrm{MI}}}{\boldsymbol{\mathrm{MN}}} \\ $$$$\boldsymbol{\mathrm{MN}}×\boldsymbol{\mathrm{PI}}+\boldsymbol{\mathrm{MN}}.\boldsymbol{\mathrm{MI}}\left(\mathrm{cos}\:\mathrm{57}+\mathrm{sin}\:\mathrm{57tan}\:\boldsymbol{\mathrm{x}}\right) \\ $$$$=\boldsymbol{\mathrm{MI}}.\boldsymbol{\mathrm{NJ}}+\boldsymbol{\mathrm{MI}}.\boldsymbol{\mathrm{MN}}\left(\mathrm{cos}\:\mathrm{57}+\mathrm{sin}\:\mathrm{57tan}\:\boldsymbol{\mathrm{x}}\right) \\ $$$$ \\ $$$$\boldsymbol{\mathrm{MN}}.\boldsymbol{\mathrm{PI}}=\boldsymbol{\mathrm{MI}}.\boldsymbol{\mathrm{NJ}}\:\:?\frac{\boldsymbol{\mathrm{MI}}}{\boldsymbol{\mathrm{MN}}}=\frac{\boldsymbol{\mathrm{PI}}}{\boldsymbol{\mathrm{NJ}}}=\frac{\boldsymbol{\mathrm{MP}}}{\boldsymbol{\mathrm{MJ}}} \\ $$$$\boldsymbol{\mathrm{MI}}=\frac{\boldsymbol{\mathrm{MN}}×\boldsymbol{\mathrm{PI}}}{\boldsymbol{\mathrm{NJ}}} \\ $$$$ \\ $$$$\frac{\mathrm{sin}\:\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{MI}}}=\mathrm{sin}\:\boldsymbol{\mathrm{x}}\left(\:\frac{\mathrm{sin}\:\left(\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos}\:\mathrm{57}\right)}{\boldsymbol{\mathrm{PI}}}\right) \\ $$$$\frac{\mathrm{1}}{\boldsymbol{\mathrm{MI}}}=\frac{\left.\mathrm{sin}\:\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos}\:\mathrm{57}\right)}{\boldsymbol{\mathrm{PI}}} \\ $$$$\boldsymbol{\mathrm{MI}}/\boldsymbol{\mathrm{PI}}=\boldsymbol{\mathrm{MN}}/\boldsymbol{\mathrm{NJ}}= \\ $$$$\boldsymbol{\mathrm{MI}}^{\mathrm{2}} +\boldsymbol{\mathrm{PI}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{\mathrm{MI}}.\boldsymbol{\mathrm{PI}}\mathrm{cos}\:\mathrm{123}=\boldsymbol{\mathrm{MP}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{MI}}^{\mathrm{2}} +\boldsymbol{\mathrm{MI}}^{\mathrm{2}} \left(\mathrm{sin}\:\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos}\:\mathrm{57}\right) \\ $$$$−\mathrm{2}.\boldsymbol{\mathrm{MI}}^{\mathrm{2}} .\left(\mathrm{sin}\:\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos}\:\mathrm{57}\right)\mathrm{cos}\:\mathrm{123}=\boldsymbol{\mathrm{MP}}^{\mathrm{2}} \\ $$$$ \\ $$$$=\boldsymbol{\mathrm{MI}}^{\mathrm{2}} \left[\mathrm{1}+\left(\mathrm{sin57}/\boldsymbol{\mathrm{t}}\:−\mathrm{cos}\:\mathrm{57}\right)−\mathrm{2}\left(\mathrm{sin}\:\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos57}\:\right)\mathrm{cos}\:\mathrm{123}\right. \\ $$$$\boldsymbol{\mathrm{MI}}^{\mathrm{2}} \left[\left(\mathrm{1}+\left(\mathrm{sin57}/\boldsymbol{\mathrm{t}}\:−\mathrm{cos57}\:\right)\left(\mathrm{1}−\mathrm{2cos}\:\mathrm{123}\right)\right]=\boldsymbol{\mathrm{MP}}^{\mathrm{2}} \right. \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\left(\frac{\boldsymbol{\mathrm{MP}}}{\boldsymbol{\mathrm{MI}}}\right)^{\mathrm{2}} =\left[\mathrm{1}+\left(\mathrm{sin}\:\mathrm{57}/\boldsymbol{\mathrm{t}}−\mathrm{cos}\:\mathrm{57}\right)\left(\mathrm{1}−\mathrm{cos}\:\mathrm{123}\right)\right] \\ $$$$\:\frac{\mathrm{sin}\:\mathrm{123}}{\boldsymbol{\mathrm{MP}}}=\frac{\mathrm{sin}\:\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{MI}}}\:\Rightarrow\:\:\:\:\frac{\boldsymbol{\mathrm{MP}}}{\boldsymbol{\mathrm{MI}}}=\frac{\mathrm{sin}\:\mathrm{123}}{\mathrm{sin}\:\mathrm{x}} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{donc}} \\ $$$$\:\:\:\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}}{\mathrm{sin}\:^{\mathrm{2}} \boldsymbol{\mathrm{x}}}=\left(\mathrm{1}+\frac{\mathrm{sin}\:\mathrm{57}}{\boldsymbol{\mathrm{t}}}−\mathrm{cos}\:\mathrm{57}\right)\left(\mathrm{1}−\mathrm{cos}\:\mathrm{123}\right) \\ $$$$\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \boldsymbol{\mathrm{x}}}=\mathrm{1}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{t}}^{\mathrm{2}} } \\ $$$$ \\ $$$$\:\:\:\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}\left(\mathrm{1}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} \right)}{\boldsymbol{\mathrm{t}}^{\mathrm{2}} }=\left(\mathrm{1}−\mathrm{cos}\:\mathrm{223}\right)\left(\frac{\mathrm{sin}\:\mathrm{57}+\boldsymbol{\mathrm{t}}−\boldsymbol{\mathrm{t}}\mathrm{cos}\:\mathrm{57}}{\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\right)\boldsymbol{\mathrm{t}} \\ $$$$ \\ $$$$\left(\mathrm{1}−\mathrm{cos}\:\mathrm{223}\right)\left(\boldsymbol{\mathrm{t}}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\mathrm{57}\right)+\mathrm{sin}\:\mathrm{57}\boldsymbol{\mathrm{t}}\right. \\ $$$$−\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}×\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}=\mathrm{0} \\ $$$$\left(\mathrm{1}−\mathrm{cos}\:\mathrm{57}\right)\left(\mathrm{1}−\mathrm{cos}\:\mathrm{123}\right)\boldsymbol{\mathrm{t}}^{\mathrm{2}} +\mathrm{sin}\:\mathrm{57}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{223}\right)\boldsymbol{\mathrm{t}} \\ $$$$−\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{sin}\:^{\mathrm{2}} \mathrm{123}=\mathrm{0} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{l}}\:\boldsymbol{\mathrm{equation}}\:\boldsymbol{\mathrm{finale}}\:\boldsymbol{\mathrm{est}}: \\ $$$$\left.\left[\left(\mathrm{1}−\boldsymbol{\mathrm{Cos}}\:\mathrm{57}\right)\left(\mathrm{1}−\boldsymbol{\mathrm{cos}}\:\mathrm{123}\right)−\boldsymbol{\mathrm{sin}}\:^{\mathrm{2}} \mathrm{123}\right)\right]\boldsymbol{\mathrm{t}}^{\mathrm{2}} \\ $$$$+\boldsymbol{\mathrm{sin}}\:\mathrm{57}\left(\mathrm{1}−\boldsymbol{\mathrm{cos}}\:\mathrm{123}\right)\boldsymbol{\mathrm{t}}−\boldsymbol{\mathrm{sin}}\:^{\mathrm{2}} \mathrm{123}=\mathrm{0} \\ $$$$ \\ $$$$\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{57}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{57}\right)\boldsymbol{\mathrm{t}}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\left(\mathrm{sin}\:\mathrm{57}+\mathrm{sin}\:\mathrm{57cos}\:\mathrm{57}\right)\boldsymbol{\mathrm{t}}=\mathrm{sin}\:^{\mathrm{2}} \mathrm{57} \\ $$$$\boldsymbol{\mathrm{soit}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{t}}=\frac{\mathrm{sin}\:\mathrm{57}}{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{57}\right)} \\ $$$$\:\:\boldsymbol{\mathrm{x}}=\mathrm{28},\mathrm{5} \\ $$$$\Rightarrow\:\:\begin{cases}{\measuredangle\boldsymbol{\mathrm{IBN}}=\mathrm{57}+\boldsymbol{\mathrm{x}}=\mathrm{85},\mathrm{5}}\\{\measuredangle\boldsymbol{\mathrm{JMP}}=\mathrm{57}−\boldsymbol{\mathrm{x}}=\mathrm{28},\mathrm{5}°}\end{cases} \\ $$$$ \\ $$
Commented by a.lgnaoui last updated on 08/Feb/25

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