Machinery's Handbook, 31st Edition
Spherical Trigonometry 119 The side and angle labels in the examples that follow refer to those of the oblique spher- ical triangle in Fig. 2. Example: An oblique spherical triangle is to be constructed on the surface of a sphere of unknown size. The length of side S will be 5.470 inches; the spherical angle of arc S must be 51 ° 17 ′ 31 ″ (angle R in Fig. 2). Angle D must be 59 ° 55 ′ 10 ″ , and angle E must be 85 ° 36 ′ 32 ″ . Find the size of the sphere, lengths of sides B and C , and the value of angle N . Solution: Convert known angles to decimal degree format to simplify calculations: R 51 ° 51.291944 ° = =
17 60 --- 31 3600 ------ + + 10 3600 ------ + + 32 3600 ------ + + 55 60 --- 36 60 ---
59 °
59.919444 °
=
=
D
85 °
85.608889 °
=
=
E
Find the radius of the sphere:
180 π -----
180 π -----
5.470 51.291944 ° -------------- ×
S R ° --- ×
L = Find the values of angles of G and H in order to get lengths of sides B and C . Then solve for the value of angle N , and finally the area. Remember that both J and L are radii, thus J = L . G sin R sin E sin csc D × × 0.780342 0.997065 × 1.15564 × = = = 0.899148 G sin –1 0.899148 ( ) 64.046301 ° = = B J π 180 ----- G × × ° 6.11 π 180 ----- × 64.046301 ° × 6.829873 inches = = = 6.11 inches = =
D E + 2 ------- D E – 2 ------- sin -------------- sin
R G – 2 -------
72.76417 sin –12.844723 ( ) sin --------------------- ( )
H 2 -- tan
=
=
–6.377185 ( ) tan ×
tan ×
0.955093 –0.222310 ------------- 0.111765 – ( ) = = 0.480167 H 2 -- tan –1 0.480167 ( ) 25.648772 ° = = ,
H 51.297543 ° =
π 180 ----- H × × ° 6.11 =
π 180 ----- 51.297543 ° × × 5.470350 inches =
C J =
R G + 2 ------- R G – 2 ------- sin -------------- sin
D E – 2 -------
57.669123 ( ) sin –6.377185 ( ) sin -------------------
N 2 -- cot
=
=
–12.844723 ( ) tan ×
tan ×
0.844974 –0.111073 ------------- 0.228015 – ( ) = = 1.7345957 N 2 -- cot –1 1.7345957 ( ) 29.963587 ° = = ,
N 59.927175 ° =
L 2 π 180 ----- D E N 180 ° – + + ( ) × 16.585 in 2 = =
Area
The triangle is an isosceles spherical triangle with legs B and C each being 5.470 inches. If angle E 1 or E 2 is known, then any problem involving oblique spherical triangles can be solved as two right spherical triangles; in that case, the equations for right spherical triangles are used.
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