Answer to Question #138994 in Trigonometry for jay nerves

Question #138994
Find the missing parts of an isosceles spherical triangle.
1.) 6. a = b = 78°20’’ C = 118°50’
2.) A = B = 95°5’ C = 100°10’
3.) B = 72°48’ b = 64°52’
4.) A = C = 50°10’ c = 95°
5.) B = C = 78°44’ b = 18°16’
1
Expert's answer
2020-10-21T14:20:04-0400

1.) 6. a = b = 78°20’’ C = 118°50’

Cosine Law for sides


cosc=cosacosb+sinasinbcosC\cos c=\cos a\cos b+\sin a\sin b\cos C

cosc=cos2a+sin2acosC\cos c=\cos^2 a+\sin^2 a\cos C

cosc=cos278°20+sin278°20cos118°500.4182\cos c=\cos^2 78\degree20''+\sin^2 78\degree20''\cos 118\degree50'\approx-0.4182

c114°4326c\approx114\degree43'26''

Sine Law

sinasinA=sinbsinB=sincsinC\dfrac{\sin a}{\sin A}=\dfrac{\sin b}{\sin B}=\dfrac{\sin c}{\sin C}

sinA=sinB=sinasinCsinc\sin A=\sin B=\dfrac{\sin a\sin C}{\sin c}

sinA=sinB=sin78°20sin118°50sin114°4326\sin A=\sin B=\dfrac{\sin 78\degree20''\sin 118\degree50'}{\sin 114\degree43'26''}

A=B=70°3735A=B=70\degree37'35''

Or


A=B=109°2225A=B=109\degree22'25''

2.) A = B = 95°5’ C = 100°10’

Cosine Law for angles


cosA=cosBcosC+sinBsinCcosa\cos A=-\cos B\cos C+\sin B\sin C\cos a

cosB=cosAcosC+sinAsinCcosb\cos B=-\cos A\cos C+\sin A\sin C\cos b

cosC=cosAcosB+sinAsinBcosc\cos C=-\cos A\cos B+\sin A\sin B\cos c

a=cos1cos(95°5)+cos(95°5)cos(100°10)sin(95°5)sin(100°10)a=\cos^{-1}\dfrac{\cos(95\degree5')+\cos(95\degree5')\cos(100\degree10')}{\sin(95\degree5')\sin(100\degree10')}

a94°165a\approx94\degree16'5''

b94°165b\approx94\degree16'5''

c=cos1cos(100°10)+cos2(95°5)sin2(95°5)c=\cos^{-1}\dfrac{\cos(100\degree10')+\cos^2(95\degree5')}{\sin^2(95\degree5')}

c99°4715c\approx99\degree47'15''

3.) A=B = 72°48’ b = 64°52’


a=b=64°52a=b=64\degree52'

Napier's analogies


cos(12(AB))cos(12(A+B))=tan(12(a+b))tan(12c)\dfrac{\cos (\dfrac{1}{2}(A-B))}{\cos (\dfrac{1}{2}(A+B))}=\dfrac{\tan (\dfrac{1}{2}(a+b))}{\tan (\dfrac{1}{2}c)}

tan(12c)=cosAtana\tan (\dfrac{1}{2}c)=\cos A\tan a

c=2tan1(cos(72°48)tan(64°52))64°2651c=2\tan^{-1}(\cos (72\degree48')\tan (64\degree52'))\approx64\degree26'51''

Sine Law

sinasinA=sinbsinB=sincsinC\dfrac{\sin a}{\sin A}=\dfrac{\sin b}{\sin B}=\dfrac{\sin c}{\sin C}

sinC=sincsinBsinb\sin C=\dfrac{\sin c\sin B}{\sin b}


C=sin1(sin(72°48)sin(64°2651)sin(64°52))C=\sin^{-1}(\dfrac{\sin (72\degree48')\sin (64\degree26'51'')}{\sin (64\degree52')})

C72°1015C\approx72\degree10'15''

Or


C107°4945C\approx107\degree49'45''

4.) A = C = 50°10’ c = 95°

a=c=95°a=c=95\degree

Napier's analogies


cos(12(AC))cos(12(A+C))=tan(12(a+c))tan(12b)\dfrac{\cos (\dfrac{1}{2}(A-C))}{\cos (\dfrac{1}{2}(A+C))}=\dfrac{\tan (\dfrac{1}{2}(a+c))}{\tan (\dfrac{1}{2}b)}

tan(12b)=cosAtana\tan (\dfrac{1}{2}b)=\cos A\tan a

tan(12b)=cos(52°10)tan(95°)7.0108\tan (\dfrac{1}{2}b)=\cos (52\degree10')\tan (95\degree)\approx-7.0108

12b>90°=>b>180°{1\over 2}b>90\degree=>b>180\degree

a+c+b>95°+95°+180°=370°a+c+b>95\degree+95\degree+180\degree=370\degree

The sum of the three sides of a spherical triangle is less than 360°.360\degree. 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°16c=b=18\degree16'

Napier's analogies


cos(12(BC))cos(12(B+C))=tan(12(b+c))tan(12a)\dfrac{\cos (\dfrac{1}{2}(B-C))}{\cos (\dfrac{1}{2}(B+C))}=\dfrac{\tan (\dfrac{1}{2}(b+c))}{\tan (\dfrac{1}{2}a)}

tan(12a)=cosBtanb\tan (\dfrac{1}{2}a)=\cos B\tan b

a=2tan1(cos(78°44)tan(18°16))7°2247a=2\tan^{-1}(\cos (78\degree44')\tan (18\degree16'))\approx7\degree22'47''

Sine Law

sinasinA=sinbsinB=sincsinC\dfrac{\sin a}{\sin A}=\dfrac{\sin b}{\sin B}=\dfrac{\sin c}{\sin C}

sinA=sinasinBsinb\sin A=\dfrac{\sin a\sin B}{\sin b}


A=sin1(sin(78°44)sin(7°2251)sin(18°16))A=\sin^{-1}(\dfrac{\sin (78\degree44')\sin (7\degree22'51'')}{\sin (18\degree16')})

A23°121A\approx23\degree12'1''

Or


A156°4759A\approx156\degree47'59''


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment