TRIGONOMETRIYaI SFERAVI

TRIGONOMETRIYaI SFERAVI, yak sohai trigonometriyaro guyand, ki vobastagii bayni kunjho va tarafhoi sekunjahoi sferaviro meomuzad. Agar A, V, S — kunjho va a, b, s — tarafhoi muqobili in kunjho dar sekunjai sferavii AVS (nigared rasm) boshand, ongoh kunjho va tarafhoi sekunjai sferaviro az rui formulahoi zerin hisob kardan lozim ast:

=  =  , (1) cos a = cos b cos c + sin b sin c cos A, (2) cos A = -cos B cos C +sin B sin C cos a , (2₁) sin a cos B =cos b sin c – sin b cos c cos A, (3) sin A cos b = cos B sin c + sin B cos C cos a;(3₁)

dar in formulaho tarafhoi a, b, s kunjhoi markazii muvofiq chen karda meshavand va darozii onho muvofiqan ba aR, bR, sR (R — radiusi sfera) barobarand.

Az rui formulahoi trigonometriyai sferavi se elementi dilkhohi sekunjai sferaviro donista, se elementi boqimondai on, yane halli sekunjaro yoftan mumkin ast. Baroi sekunjahoi rostkunjai sferavi (A = 90°, a — gipotenuza, b, s — katetho) formulahoi trigonometriyai sferavi nisbatan soddaand:

sin b = sin a sin B, (1′)

cos a = cos b cos c, (2′)

sin a  cos B cos=cos b· sin c. (3′)

Baroi hosil kardani formulahoe, ki vobastagii elementhoi sekunjai rostkunjai sferaviro ifoda mekunand, az qoidai mnemoni (qoidan Neper) istifoda burdan mumkin ast: agar katethoi sekunjai rostkunjai sferaviro bo imtidodashon ivaz namuda, elementhoi sekunjaro (gayr az kunji rosti A) az rui davra bo hamon tartibi dar sekunja joygirshuda (yane ba tartibi V, a, S, 90° — b, 90° —c) joy dihem, on goh kosinusi har yak element ba hosili zarbi sinushoi elementhoi gayrihamsoya barobar meshavand, masalan, sos a = sin (90° — s) sin (90° — b) yo badi tabdil dodan sos a = sos b sos s (formulai 2′). Istifodai formulahoi Delambr baroi halli masalaho muvofiq ast, ki vobastagii hamai shash elementi sekunjai sferaviro ifoda mekunad:

sin  a cos  (B – C) = sin  A sin  (b+c),

sin  a sin  (B – C) = cos  A sin  (b-c) ,

cos  A cos   (B+C) = sin cos (b+c),

cos a sin (B+C) = cos  A cos  (b-c).

Hangomi halli masalaho vobasta ba sahehiyati      hal va formulahoi taqribi niz istifoda burdai mumkin ast.

Trigonometriyai sferavi nisbat ba trigonomstriyai hamvor peshtar ba vujud omadaast. Khosiyathoi sekunjai sferavi va mavridhoi gunoguni halli sekunjahoi sferavi ba matematikoni Yunon Menelay (asri 1) va Ptolemey (asri 2) malum budand. Onho halli sekunjahoi kajkunjai sferaviro ba halli sekunjahoi rostkunjai sferavi meovardand. Olim va riyozidoni buzurgi Sharq Nasiruddini Tusi hamai mavridhoi halli sekunjahoi kajkunjaro muttasil muoina karda, bori avval halli du mavridi dushvori onro nishon dod. Formulahoi asosii sekunjai kajkunjai sferaviro olimi Sharq Abulvafoi Buzajoni [formulai (1)] matematiki nemis I. Regiomontan (miyonai asri 15) [formulai navi (2)] va L. Eyder (Rossiya, asri 18) [formulai navi (3) va (3₁)] kashf kardaand. Dar inkishofi minbadai trigonometriyai sferavi matematiki shotlandi J. Neper, matematiki anglis G. Brigs, astronomi rus A. I. Leksel, astronomi fransavi J. Delambr va digaron niz sahmi kalon guzoshtaand.  A. Qurbonov.