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Question Number 106990 by I want to learn more last updated on 08/Aug/20

$$\mathrm{Two}\mathrm{places}\mathrm{A}\mathrm{and}\mathrm{B}\mathrm{both}\mathrm{on}\mathrm{a}\mathrm{parallel}\mathrm{of}\mathrm{latitude}\alpha °\mathrm{N}$$$$\mathrm{differs}\mathrm{in}\mathrm{longitudes}\mathrm{by}\theta °.\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{shortest}\mathrm{distance}$$$$\mathrm{between}\mathrm{them}\mathrm{is}:\frac{\left[2{\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)\right]}{360}\times 2\pi \mathrm{R}$$$$$$$$\mathrm{Topic}:\mathrm{Longitude}\mathrm{and}\mathrm{Latitude}$$

Commented by I want to learn more last updated on 08/Aug/20

$$\mathrm{I}\mathrm{have}\mathrm{change}\mathrm{the}\mathrm{question}\mathrm{sir}$$

Commented by I want to learn more last updated on 08/Aug/20

$$\mathrm{Ok}\mathrm{sir},\mathrm{i}\mathrm{understand}.$$$$$$$$\mathrm{Prove}\mathrm{that}\mathrm{shortest}\mathrm{distance}\mathrm{between}\mathrm{two}\mathrm{points}$$$$=\frac{\left[2{\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)\right]}{360}\times 2\pi \mathrm{R}$$$$\mathrm{Topic}:\mathrm{Longitude}\mathrm{and}\mathrm{latitude}.$$

Commented by I want to learn more last updated on 08/Aug/20

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}.\mathrm{Please}\mathrm{help}\mathrm{me}\mathrm{see}\mathrm{to}\mathrm{it}\mathrm{when}\mathrm{you}\mathrm{are}\mathrm{chanced}\mathrm{sir}$$

Commented by Tawa11 last updated on 15/Sep/21

$$\mathrm{nice}$$

Answered by mr W last updated on 08/Aug/20

Commented by mr W last updated on 08/Aug/20

$$OA=OB=R$$$$CA=CB=R\cos \alpha$$$$AB=2\times CA\times \sin \frac{\theta }{2}=2R\cos \alpha \sin \frac{\theta }{2}$$$$AB=2\times OA\times \sin \frac{\phi }{2}=2R\sin \frac{\phi }{2}$$$$\Rightarrow \sin \frac{\phi }{2}=\cos \alpha \sin \frac{\theta }{2}$$$$\Rightarrow \phi =2{\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)$$$$theshortestdistancefromAtoB$$$$onthespheresurfaceisthegreat$$$$circlearc\stackrel{⌢}{AB}:$$$$\stackrel{⌢}{AB}=\phi R=2R{\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)$$$$if\phi ={\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)isnotinrad,$$$$butindegree,then$$$$\stackrel{⌢}{AB}=2R{\sin }^{-1}\left(\cos \alpha \sin \frac{\theta }{2}\right)\times \frac{2\pi }{360°}$$

Commented by peter frank last updated on 08/Aug/20

$$\mathrm{thank}\mathrm{you}$$

Commented by I want to learn more last updated on 08/Aug/20

$$\mathrm{Wow},\mathrm{I}\mathrm{really}\mathrm{appreciate}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by peter frank last updated on 08/Aug/20

$$\mathrm{Mr}\mathrm{w}\mathrm{help}\mathrm{please}\mathrm{qn}107073$$$$\mathrm{ezz}$$

Commented by mr W last updated on 08/Aug/20

$$iwouldificould...$$

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