Two-places-A-and-B-both-on-a-parallel-of-latitude-N-differs-in-longitudes-by-Show-that-the-shortest-distance-between-them-is-2-sin-1-cos-sin-2-360-2piR-Topic-

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

Two places A and B both on a parallel of latitude α ° N

differs in longitudes by θ ° . Show that the shortest distance

between them is : \frac{[2 \sin^{- 1} (\cos α \sin \frac{θ}{2})]}{360} \times 2 π R

Topic : Longitude and Latitude

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

I have change the question sir

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

Ok sir, i understand .

Prove that shortest distance between two points

= \frac{[2 \sin^{- 1} (\cos α \sin \frac{θ}{2})]}{360} \times 2 π R

Topic : Longitude and latitude .

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

Thank you sir . Please help me see to it when you are chanced sir

Commented by Tawa11 last updated on 15/Sep/21

nice

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

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

O A = O B = R

C A = C B = R \cos α

A B = 2 \times C A \times \sin \frac{θ}{2} = 2 R \cos α \sin \frac{θ}{2}

A B = 2 \times O A \times \sin \frac{φ}{2} = 2 R \sin \frac{φ}{2}

\Rightarrow \sin \frac{φ}{2} = \cos α \sin \frac{θ}{2}

\Rightarrow φ = 2 \sin^{- 1} (\cos α \sin \frac{θ}{2})

t h e s h o r t e s t d i s t a n c e f r o m A t o B

o n t h e s p h e r e s u r f a c e i s t h e g r e a t

c i r c l e a r c \overset{⌢}{A B} :

\overset{⌢}{A B} = φ R = 2 R \sin^{- 1} (\cos α \sin \frac{θ}{2})

i f φ = \sin^{- 1} (\cos α \sin \frac{θ}{2}) i s n o t i n r a d,

b u t i n d e g r e e, t h e n

\overset{⌢}{A B} = 2 R \sin^{- 1} (\cos α \sin \frac{θ}{2}) \times \frac{2 π}{360 °}

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

thank you

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

Wow, I really appreciate sir . God bless you sir .

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

Mr w help please qn 107073

ezz

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

i w o u l d i f i c o u l d \dots

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