1.) 6. a = b = 78°20’’ C = 118°50’
Cosine Law for sides
cosc=cosacosb+sinasinbcosC
cosc=cos2a+sin2acosC
cosc=cos278°20′′+sin278°20′′cos118°50′≈−0.4182
c≈114°43′26′′Sine Law
sinAsina=sinBsinb=sinCsinc
sinA=sinB=sincsinasinC
sinA=sinB=sin114°43′26′′sin78°20′′sin118°50′
A=B=70°37′35′′ Or
A=B=109°22′25′′ 2.) A = B = 95°5’ C = 100°10’
Cosine Law for angles
cosA=−cosBcosC+sinBsinCcosa
cosB=−cosAcosC+sinAsinCcosb
cosC=−cosAcosB+sinAsinBcosc
a=cos−1sin(95°5′)sin(100°10′)cos(95°5′)+cos(95°5′)cos(100°10′)
a≈94°16′5′′
b≈94°16′5′′
c=cos−1sin2(95°5′)cos(100°10′)+cos2(95°5′)
c≈99°47′15′′ 3.) A=B = 72°48’ b = 64°52’
a=b=64°52′ Napier's analogies
cos(21(A+B))cos(21(A−B))=tan(21c)tan(21(a+b))
tan(21c)=cosAtana
c=2tan−1(cos(72°48′)tan(64°52′))≈64°26′51′′ Sine Law
sinAsina=sinBsinb=sinCsinc
sinC=sinbsincsinB
C=sin−1(sin(64°52′)sin(72°48′)sin(64°26′51′′))
C≈72°10′15′′ Or
C≈107°49′45′′
4.) A = C = 50°10’ c = 95°
a=c=95°
Napier's analogies
cos(21(A+C))cos(21(A−C))=tan(21b)tan(21(a+c))
tan(21b)=cosAtana
tan(21b)=cos(52°10′)tan(95°)≈−7.0108
21b>90°=>b>180°
a+c+b>95°+95°+180°=370° The sum of the three sides of a spherical triangle is less than 360°. Therefore the isosceles spherical triangle with A = C = 50°10’ c = 95° does not exist.
5.) B = C = 78°44’ b = 18°16’
c=b=18°16′
Napier's analogies
cos(21(B+C))cos(21(B−C))=tan(21a)tan(21(b+c))
tan(21a)=cosBtanb
a=2tan−1(cos(78°44′)tan(18°16′))≈7°22′47′′ Sine Law
sinAsina=sinBsinb=sinCsinc
sinA=sinbsinasinB
A=sin−1(sin(18°16′)sin(78°44′)sin(7°22′51′′))
A≈23°12′1′′ Or
A≈156°47′59′′
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