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SPHERICAL
TRIGONOMETRY
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SPHERICA TRIANGE
B
AC
O
b
a
O c
B
A
C
A spherical triangle is that part of the surfaceof a sphere bounded by three arcs of greatcircles. The bounding arcs are called the sides
of the spherical triangle and the intersection ofthese arcs are called the vertices. The angleformed by two intersecting arcs is called aspherical angle.
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Important Propo!ition! "rom Soli#Geometry
•If two sides are eual! the angles opposite areeual and conversely.
• If two sides are uneual ! the angles opposite areuneual and the greater side is opposite the
greater angle and conversely.• The sum of any two sides is greater than the thirdside! that is!
a " b # c! a " c # b! b " c # a• The sum of the three sides is less than $%&&! that is!
&& ' a " b " c ' $%&& • The sum of any two angles is less than ()&& plus
the third angle! that is!A " B ' ()&& " C! A " C ' ()&& " B! B " C
' ()&&
" A• The sum of the an les is reater than ()&& and less
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RIGHT SPHERICATRIANGE
a c
b
B
AC
a
bco , A
co , cco , B
-here co – A / complement of A
co – B / complement of B co – c / complement of c
Napier$! Circle
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NAPIER$S R%E01( The sine of any middle part is
eual to the product of the
tangents of the ad2acent parts.013 The sine of any middle part is
eual to the product of the
cosines of the opposite parts.a&! o" '(a#rant!
45( Any side and its opposite angle lie in thesame uadrant and conversely.
453 6a7 If any two sides lie in the same uadrant!then the third side is less than 8&& andconversely.
6b7 If any two sides lie in di9erent
uadrants! then the third side is greaterthan 8&& and conversely.
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F%NDAMENTAFORM%AS:
• tan a / tan c cos B• tan a / sin b tan A• tan b / sin a tan B• tan b / tan c cos A• cos c / cot A cot B• sin a / sin c sin A•
sin b / sin c sin B• cos c / cos a cos b• cos A / cos a sin B• cos B / cos b sin A
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E)AMPE::olve the following right spherical triangle 6C
/ 8&
&
71. a / *&&3&; c / )&&$&;2. a / +*&*
