Question #139003

Find the remaining parts of a quadrantal triangle (c = 90°).

1.) a = 70°10’ b = 52°40’

2.) a = 116°53’ A = 122°39’

3.) b = 69°29.7’ B = 63°4.6’

4.) a = 106°38’ b = 36°49’

5.) A = 52°55’ b = 73°11’


1
Expert's answer
2020-10-22T18:02:14-0400

In case giving a spherical triangle in which one side is 90° , we use the following fundamental formuleas




sinmiddle=tan(adj.)×tan(adj.)\sin middle=\tan(adj.)\times\tan(adj.)sinmiddle=cos(opp.)×cos(opp.)\sin middle=\cos(opp.)\times\cos(opp.)

1.) a = 70°10’ b = 52°40’ c = 90°


sin((90°C))=tan(90°b)×tan(90°a)\sin (-(90\degree-C))=\tan(90\degree- b)\times \tan(90\degree-a)cosC=cota×cotb\cos C=-\cot a\times\cot bC=cos1(cot(70°10)cot(52°40)105°584C=\cos^{-1}(-\cot(70\degree10')\cot(52\degree40')\approx105\degree58'4''sin(90°b)=cosB×cos(90°a)\sin (90\degree-b)=\cos B\times \cos(90\degree-a)cosB=cosbsina\cos B=\dfrac{\cos b}{\sin a}B=cos1(cos52°40sin70°10)49°5127B=\cos^{-1}(\dfrac{\cos 52\degree40'}{\sin 70\degree10'})\approx49\degree51'27''sin(90°a)=cosA×cos(90°b)\sin (90\degree-a)=\cos A\times \cos(90\degree-b)cosA=cosasinb\cos A=\dfrac{\cos a}{\sin b}A=cos1(cos70°10sin52°40)64°4428A=\cos^{-1}(\dfrac{\cos 70\degree10'}{\sin 52\degree40'})\approx64\degree44'28''

2.) a = 116°53’ A = 122°39’ c = 90°



sin(90°a)=cosA×cos(90°b)\sin (90\degree-a)=\cos A\times \cos(90\degree-b)sinb=cosacosA\sin b=\dfrac{\cos a}{\cos A}b=sin1(cos116°53cos122°39)56°5635b=\sin^{-1}(\dfrac{\cos 116\degree53'}{\cos 122\degree39'})\approx56\degree56'35''




sinB=tanA×tan(90°a)\sin B=\tan A\times \tan (90\degree-a)B=sin1(tan122°39tan116°53 )52°1750B=\sin^{-1}(\dfrac{\tan 122\degree39'}{\tan116\degree53'}\ )\approx52\degree17'50''sin((90°C))=tan(90°b)×tan(90°a)\sin (-(90\degree-C))=\tan (90\degree-b)\times \tan (90\degree-a)cosC=1tana×tanb\cos C=-\dfrac{1}{\tan a\times\tan b}C=cos1(1tan116°53×tan122°39)108°5721C=\cos^{-1}(-\dfrac{1}{\tan 116\degree53'\times\tan 122\degree39'})\approx108\degree57'21''


3.) b = 69°29.7’ B = 63°4.6’ c = 90°



sinA=tanB×tan(90°b)\sin A=\tan B\times \tan (90\degree-b)A=sin1(tan63°4.6tan69°29.7)47°25.6A=\sin^{-1}(\dfrac{\tan63\degree4.6'}{\tan69\degree29.7'})\approx47\degree25.6'sin(90°b)=cosB×cos(90°a)\sin (90\degree-b)=\cos B\times \cos(90\degree-a)sina=cosbcosB\sin a=\dfrac{\cos b}{\cos B}a=sin1(cos69°29.7cos63°4.6)50°40.8a=\sin^{-1}(\dfrac{\cos69\degree29.7'}{\cos63\degree4.6'})\approx50\degree40.8'sinB=cos(90°b)×cos((90°C))\sin B=\cos (90\degree-b)\times \cos(-(90\degree-C))sinC=sinBsinb\sin C=\dfrac{\sin B}{\sin b}C=sin1(sin63°4.6sin69°29.7)72°9.7C=\sin^{-1}(\dfrac{\sin63\degree4.6'}{\sin69\degree29.7'})\approx72\degree9.7'

4.) a = 106°38’ b = 36°49’ c = 90°



sin((90°C))=tan(90°b)×tan(90°a)\sin (-(90\degree-C))=\tan(90\degree- b)\times \tan(90\degree-a)cosC=cota×cotb\cos C=-\cot a\times\cot bC=cos1(cot(106°38)cot(36°49)66°2841C=\cos^{-1}(-\cot(106\degree38')\cot(36\degree49')\approx66\degree28'41''sin(90°b)=cosB×cos(90°a)\sin (90\degree-b)=\cos B\times \cos(90\degree-a)cosB=cosbsina\cos B=\dfrac{\cos b}{\sin a}B=cos1(cos36°49sin106°38)49°5127B=\cos^{-1}(\dfrac{\cos 36\degree49'}{\sin 106\degree38'})\approx49\degree51'27''sin(90°a)=cosA×cos(90°b)\sin (90\degree-a)=\cos A\times \cos(90\degree-b)cosA=cosasinb\cos A=\dfrac{\cos a}{\sin b}A=cos1(cos106°38sin36°49)118°32A=\cos^{-1}(\dfrac{\cos 106\degree38'}{\sin 36\degree49'})\approx118\degree32'



5.) A = 52°55’ b = 73°11’



sinA=tanB×tan(90°b)\sin A=\tan B\times \tan (90\degree-b)tanB=sinA×tanb\tan B=\sin A\times\tan bB=tan1(sin52°55×tan73°11)69°152B=\tan^{-1}(\sin52\degree55'\times\tan73\degree11')\approx69\degree15'2''sin(90°a)=cosA×cos(90°b)\sin (90\degree-a)=\cos A\times \cos(90\degree-b)a=cos1(cos52°55×sin73°11)54°4449a=\cos^{-1}(\cos52\degree55'\times\sin73\degree11')\approx54\degree44'49'sin(90°b)=tan((90°C))×tanA\sin (90\degree-b)=\tan (-(90\degree-C))\times \tan AtanC=tanAcosb\tan C=-\dfrac{\tan A}{\cos b}C=180°tan1(tan52°55cos73°11)102°205C=180\degree-\tan^{-1}(\dfrac{\tan52\degree55'}{\cos73\degree11'})\approx102\degree20'5''

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