Machinery's Handbook, 31st Edition
118 Spherical Trigonometry Oblique Spherical Trigonometry.— The heavy solid lines B , C , and S of Fig. 2 represent the sides of an oblique spherical triangle. The dashed lines J and L are radii of the sphere extending from the center of the sphere to the vertices of the triangle. The several plane triangles, indicated by the various broken lines, are formed from the radii and vertices of the spherical triangle. J and L are radii and thus have the same value.
Fig. 2. Oblique Spherical Triangle Formulas for Oblique Spherical Triangles Formulas for Lengths
π 180 ----- G × × ° =
π 180 ----- H × × ° =
π 180 ----- R × × °
180 π -----
180 π -----
B G ° ---- ×
S R ° --- ×
=
=
J
L
S L =
B J
C J
Formulas for Angles
180 π -----
180 π -----
180 π -----
B J -- ×
C J -- ×
S L -- ×
G °
H °
R °
=
=
=
Angular Relationships
Angle
Relationships
Angle
Relationships
D G N
E
E sin csc G × × E sin csc D × ×
G sin csc R × ×
D sin
=
R sin
E sin
=
D sin
E E
×
G sin
=
R sin
E 1 cot
=
D tan
H cos
1
D cos csc E 1 ×
E 2 sin ×
×
N cos
=
E 2 cot
=
N tan
R cos
2
R G + 2 ------- R G – 2 -------
D E + 2 ------- D E – 2 -------
sin
sin
D E – 2 -------
R G – 2 -------
N 2 -- cot
H 2 -- tan
N
H
tan ×
tan ×
sin --------------
sin --------------
=
=
R
G sin ×
E csc ×
R sin
=
D sin
Area Formula
π 180 ----- D E N 180 ° – + + ( ) ×
2
Area
=
L
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