The Museum of HP Calculators
The Museum of HP Calculators
This program is Copyright © 2005 by Jean-Marc Baillard and is used here by permission.
This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
-This program finds the sides and the angles of a spherical triangle.
-It works in all angular modes but angles must be entered as decimals.
-It conforms to the standard triangle notation ( A opposite a ... etc
... )
Formulae: sin a / sin A = sin
b / sin B = sin c / sin C
cos a = cos b cos c + sin b sin c cos A
cos A = -cos B cos C + sin B sin C cos a
c = arctan ( cos A tan b ) + arctan ( cos B tan a ) (
modulo 180° )
-This last formula is used in cases n°5 and n°6.
-Other formulae can be used, for example:
tan c/2 = ( tan (a-b)/2 ).( sin (A+B)/2 ) / ( sin (A-B)/2 )
( F1 )
or tan c/2 = ( tan (a+b)/2 ).( cos (A+B)/2 ) / ( cos (A-B)/2 )
( F2 )
but ( F1 ) cannot be applied if a = b & A = B and ( F2 ) doesn't work if a + b = A + B = 180°
A
*
*
* b
c *
*
with A + B + C > 180°
*
*
B *
* * * *
* * C
a
Program Listing
Data Registers: When
the program stops: R01 = a
R04 = A
( R00: temp )
R02 = b R05 =
B
R03 = c R06 =
C
Flags: /
Subroutines: /
01 LBL "SABC"
02 GTO IND T
03 LBL 00
04 STO 04
05 RDN
06 STO 05
07 X<>Y
08 STO 06
09 R^
10 XEQ 11
11 STO 01
12 LBL 07
13 RCL 04
14 RCL 06
15 RCL 05
16 XEQ 11
17 STO 02
18 RCL 04
19 RCL 05
20 RCL 06
21 XEQ 11
22 STO 03
23 RCL 02
24 RCL 01
25 RTN
26 LBL 01
27 STO 01
28 RDN
29 STO 05
30 X<>Y
31 STO 06
32 R^
33 CHS
34 XEQ 12
35 STO 04
36 XEQ 07
37 CLX
38 RCL 04
39 RTN
40 LBL 02
41 STO 01
42 X<>Y
43 STO 02
44 X<> Z
45 STO 06
46 XEQ 12
47 STO 03
48 XEQ 08
49 RCL 03
50 RCL 05
51 RCL 04
52 RTN
53 LBL 03
54 STO 01
55 X<>Y
56 STO 02
57 X<> Z
58 STO 03
59 CHS
60 XEQ 11
61 STO 06
62 LBL 08
63 RCL 01
64 RCL 03
65 RCL 02
66 CHS
67 XEQ 11
68 STO 05
69 RCL 02
70 RCL 03
71 RCL 01
72 CHS
73 XEQ 11
74 STO 04
75 RCL 06
76 RCL 05
77 RCL 04
78 RTN
79 LBL 04
80 STO 01
81 SIN
82 X<>Y
83 STO 04
84 SIN
85 /
86 X<>Y
87 STO 05
88 SIN
89 *
90 ASIN
91 LBL 09
92 STO 02
93 XEQ 14
94 RCL 02
95 FC?C 25
96 CLST
97 RTN
98 RCL 02
99 COS
100 CHS
101 ACOS
102 GTO 09
103 LBL 05
104 STO 01
105 SIN
106 X<>Y
107 STO 02
108 SIN
109 /
110 X<>Y
111 STO 05
112 SIN
113 *
114 ASIN
115 LBL 10
116 STO 04
117 XEQ 14
118 RCL 04
119 FC?C 25
120 CLST
121 RTN
122 RCL 04
123 COS
124 CHS
125 ACOS
126 GTO 10
127 LBL 11
128 XEQ 13
129 +
130 X<>Y
131 /
132 ACOS
133 RTN
134 LBL 12
135 XEQ 13
136 RCL Z
137 *
138 +
139 ACOS
140 RTN
141 LBL 13
142 STO 00
143 SIGN
144 P-R
145 X<> Z
146 1
147 P-R
148 ST* T
149 RDN
150 *
151 ABS
152 X<>Y
153 RCL 00
154 COS
155 RTN
156 LBL 14
157 SF 25
158 RCL 02
159 TAN
160 RCL 04
161 COS
162 *
163 ATAN
164 RCL 01
165 TAN
166 RCL 05
167 COS
168 *
169 ATAN
170 +
171 1
172 CHS
173 ACOS
174 MOD
175 STO 03
176 RCL 01
177 RCL 02
178 RCL 03
179 CHS
180 XEQ 11
181 STO 06
182 RCL03
183 END
( 233 bytes / SIZE 007 )
-Execution time ~ 8 seconds
1°) First case: the 3 angles are known
| STACK | INPUTS | OUTPUTS |
| T | 0 | / |
| Z | C | c |
| Y | B | b |
| X | A | a |
Example: A = 50° , B = 70° , C = 100°
0 ENTER^
100 ENTER^
70 ENTER^
50 XEQ "SABC" >>>> a = 50.9193°
RDN b = 72.2173° RDN c =
86.3204°
2°) Second case: 1 side and the 2 adjacent angles
are known
| STACK | INPUTS | OUTPUTS |
| T | 1 | / |
| Z | C | c |
| Y | B | b |
| X | a | A |
Example: a = 30° , B = 50° , C = 100°
1 ENTER^
100 ENTER^
50 ENTER^
30 XEQ "SABC" >>>> A = 40.0971°
RDN b = 36.4896° RDN c =
49.8627°
3°) Third case: 2 sides and the included angle
are known
| STACK | INPUTS | OUTPUTS |
| T | 2 | / |
| Z | C | c |
| Y | b | B |
| X | a | A |
Example: a = 30° b = 40° C = 80°
2 ENTER^
80 ENTER^
40 ENTER^
30 XEQ "SABC" >>>> A = 45.1309°
RDN B = 65.6596° RDN c =
44.0096°
4°) Fourth case: 3 known sides
| STACK | INPUTS | OUTPUTS |
| T | 3 | / |
| Z | c | C |
| Y | b | B |
| X | a | A |
Example: a = 30° b = 40° c = 50°
3 ENTER^
50 ENTER^
40 ENTER^
30 XEQ "SABC" >>>>
A = 40.6441° RDN B = 56.8634°
RDN C = 93.6796°
5°) Fifth case: 1 side , the opposite angle
and another angle are known
| STACK | INPUTS | OUTPUTS1 | OUTPUTS2 |
| T | 4 | / | / |
| Z | B | C | C' |
| Y | A | c | c' |
| X | a | b | b' |
Example: a = 30° A = 40° B = 60°
4 ENTER^
60 ENTER^
40 ENTER^
30 XEQ "SABC" >>>> b
= 42.3493° RDN c =
51.0269° RDN C = 91.8854°
R/S
>>>> b' = 137.6507° RDN
c' = 161.1774° RDN C' = 155.4947°
-Here, we have 2 solutions.
-If one of the solutions must be rejected, the stack is cleared
( X = Y = Z = T = 0 )
-For instance a = 30° b = 110° B = 40°
4 ENTER^
40 ENTER^
110 ENTER^
30 XEQ "SABC" >>>> b = 20.0000°
RDN c = 16.7627° RDN C =
32.8220°
R/S >>>>
0 RDN 0 RDN 0
-Thus , there is only one solution.
6°) Sixth case: 2 sides and one adjacent angle
known
| STACK | INPUTS | OUTPUTS1 | OUTPUTS2 |
| T | 5 | / | / |
| Z | B | C | C' |
| Y | b | c | c' |
| X | a | A | A' |
Example: a = 100° b = 110° B = 120°
5 ENTER^
120 ENTER^
110 ENTER^
100 XEQ "SABC" >>>>
A = 65.1763° RDN c = 21.4983°
RDN C = 19.7395°
R/S
>>>> A' = 114.8237° RDN c' = 119.6509°
RDN C' = 126.7814°
-In this example, there are also 2 solutions.
-If one of the solutions must be rejected, the stack is cleared
( X = Y = Z = T = 0 )
-In cases 1-2-3-4 , T-register input = the number of known sides.
-In cases 5-6 , the results are not very accurate if all angles are
near 90°
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