SPHERICAL TRIGONOMETRY Spherical Trigonometry Machinery's Handbook, 31st Edition
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Spherical trigonometry deals with the measurement of angles of polygons—triangles in particular—that lie on the surface of spheres. The sides of a spherical triangle conform to the surface of the sphere, and, unlike a plane triangle, the sum of angle measures of a spherical triangle ranges from 180 ° to 540 ° . Right-Angle Spherical Trigonometry.— The solid black lines A , B , and C of Fig. 1 rep resent the sides of a right spherical triangle. The dashed lines J and K are radii of the sphere extending from the center of the sphere to the triangle’s vertices. The several plane trian gles, indicated by the various broken lines, are formed from the radii and vertices of the spherical triangle. J and K are radii and thus have the same value.
Fig. 1. Right Spherical Triangle Formulas for Right Spherical Triangles Formulas for Lengths
π 180 ----- F × × °
π 180 ----- G × × ° =
π 180 ----- H × × ° =
180 π -----
180 π -----
B G ° ---- ×
A F ° --- ×
=
=
J
K
A K =
B J
C J
Formulas for Angles
180 π -----
180 π -----
180 π -----
A K -- ×
B J -- ×
C J -- ×
F °
G °
H °
=
=
=
Angle
Angular Relationships
D sin
F sin csc H × =
D cos E cos F cos G cos H cos
=
G tan G cos G sec H cos
H cot × D sin × H cos × F sec ×
D tan E tan F tan G tan
F tan csc G × = G tan csc F × =
D E F G H
= =
F sin
=
G tan
E cot ×
=
D tan F sin
G sin × E tan ×
=
=
H cos
G cos
D cot = Area Formula
=
F cos ×
E cot ×
π 180 ----- D ° E ° 90 ° 180 ° – + + ( ) ×
π 180 ----- D ° E ° 90 ° + – ( ) ×
2
2
Area
=
=
K
K
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