553
1 INTRODUCTION
The formulas of spherical triangle, which include
many different kinds of formulas such as the sine
rules, the cosine rules for the sides, the four‐part
formulas, the five‐part formulas, etc. are the
important basic knowledge of nautical mathematics.
They are widely applied to solve the various
navigation problems. Hsu et al. have provided an
architecture diagram to clarify the relationships of
these formulas (Hsu et al., 2005). On this basic, we
add the formulas of the right spherical triangle and
the formulas of the quadrantal spherical triangle to
reorganize the diagram as shown in Figure 1.
For
instance,thesine rulescanbederivedfromthecosine
rules for the sides. The four‐part formulas and the
five‐part formulas (I) can be derived from the sine
rulesandthecosinerulesforthesides.Thefive‐part
formulas(II)canbederivedfromthecosinerules
for
the sides or the polar duality property of five‐part
formulas(I),etc.Inaddition,thetwospecialcases,the
rightsphericaltrianglewhichoneangleis90°andthe
quadrantal spherical triangle which one side is 90°,
canbederivedfromthecosinerulesforthesides,the
sine
rules, and the four‐part formulas. As shown in
Figure 1, the cosine rules for the sides and the sine
rulesarethefundamentalformulastoderivetheother
sphericaltriangleformulas,hence,theyarealsocalled
the genetic codes of the spherical triangle formulas.
To derive these formulas, many scholars
have
proposed different approaches as follows: Clough‐
Smith(1966),Todhunter(1886)andSmart(1977)have
presented a geometric method to derive the cosine
rulesforthesidesandthesinerules,Green(1985)has
providedavector methodtoderive the cosinerules
forthesidesandthesinerules,
etc.However,inthe
teachingprocess,whentheteacherusesthegeometric
method to derive and prove the sine rules and the
cosine rules for the sides, students usually feel the
process is complicated and difficult to understand.
Derivation of Formulas in Spherical Trigonometry
Based on Rotation Matrix
T.H.Hsieh,S.Z.Wang,W.Liu&J.S.Zhao
ShanghaiMaritimeUniversity,Shanghai,China
C.L.Chen
NationalTaiwanOceanUniversity,Keelung,Taiwan
ABSTRACT:Theformulasofsphericaltriangle,whicharewidelyusedtosolvevarious navigationproblems,
aretheimportantbasicknowledgeofnauticalmathematics.Becausethesinerulesandthecosinerulesforthe
sides are thefundamentalformulasto derive the other spherical triangle formulas,
they are also called the
geneticcodesof the spherical triangleformulas. In theteaching process, teachers usuallyuse the geometric
method toderive and prove thesefundamental formulas. However, thederivationof geometric methods is
complicated and difficult to understand. To improve the teaching process, this paper proposes the
three‐
dimensional rotation method, which is based on conversion of two cartesian coordinate frames using the
rotation matrices. This method caneasily and simultaneously derive the sine rules, the cosine rules for the
sides, and the five‐part formulas (I), and is also helpful to solve different kinds of spherical navigation
problems.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 3
September 2019
DOI:10.12716/1001.13.03.09
554
Althoughusingthevectormethodinsteadmayletthe
process become simpler, different fundamental
formulas stillneedto be derived respectively. Thus,
this paper proposes the three‐dimensional rotation
methodto convertbetweentwo cartesiancoordinate
framesusingthe rotation matrices.This method can
easily and simultaneously derive the sine
rules, the
cosinerulesforthesides,andthefive‐partformulas
(I).
Figure1. Relationships of the various spherical triangle
formulas.
2 DERIVATIONOFSPHERICALTRIANGLE
FORMULAS
2.1 Twocartesiancoordinateframesaredefined
Aspherical triangle(
ABC )on anunitsphere has
six elements which include the three sides (
a , b ,
and
c
)and threeangles(
,
,and
).First, we
set up the frame A (
A
x
,
A
y
,
A
z ) and select the
pointAasapoleoftheframeAonanunitsphere,as
showninFigure2.ThecoordinatesofthepointCon
theunitsphereintheframeA(
A
C )isasfollows:



A
cos 90 sin 180
C cos 90 cos 180
sin 90
b
b
b

 






(1)
90cos
b
Figure2.LocatingthepointCintheframeA.
O
A
x
B
y
B
z
B
B
C
90cos
a
Figure3.LocatingthepointCintheframeB.
Then,wecreatetheframeB(
B
x
,
B
y
,
B
z )andset
theaxisZoftheframeBintersectsanunitsphereat
thepointB,asshowninFigure3.Thecoordinatesof
thepointContheunitsphereintheframeB(
B
C )is
asfollows:


B
cos 90 sin
C cos 90 cos
sin 90
a
a
a



(2)
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2.2 Transformationofcoordinateframesonbasicrotation
Figure4. Conversion of frame B to frame A using the
clockwiserotationmatrix.
AsshowninFigure4,whentheframeAandthe
frame B are displayed on the same figure and the
originis the same,the coordinatesof thepoint Cin
both frames (
A
C and
B
C ) represent the position of
the same point C. When the frame B is rotated
clockwisetheangleofthesidecalongtheaxis
B
x
to
overlapitwiththeframeA,thecoordinates
B
C can
also be converted to coordinates
A
C by using the
clockwiserotationmatrixasshowninEquation3.The
clockwise rotation matrix of the axis
x
is listed in
Equation4(Arfken,1985).

AB
CR C
B
xc (3)

10 0
R0cossin
0sin cos
x







(4)
Substitute Equations 1, 2 and 4 into Equation 3,
andrearrangetheformulasasfollows:
sin sin sin sinba
(5)
sin cos cos sin cos sin cosbcaca
(6)
cos sin sin cos cos cosbca ca
(7)
Andthen,asshowninFigure5,whentheframeA
is rotated counterclockwise the angle of the side c
alongtheaxis
A
x
tooverlapitwiththeframeB,the
coordinates
A
C canalsobeconvertedtocoordinates
B
C byusingthecounterclockwiserotationmatrixas
shown in Equation 8. The counterclockwise rotation
matrixoftheaxis
x
islistedinEquation9(Arfken,
1985).
BA
CR C
B
xc
(8)

10 0
R0cossin
0sincos
x

(9)
Substitute Equations 1, 2 and 9 into Equation 8,
andrearrangetheformulasasfollows:
sin sin sin sinab
(10)
sin cos cos sin cos sin cosacbcb

(11)
cos sin sin cos cos cosacb cb
(12)
Equations5and10arethesine rules;Equations6
and11arethefive‐partformulas(I);Equations7and
12arethecosinerulesforthesides.Theresultsshow
thatthethree‐dimensionalrotationmethodcanderive
the sinerules, the cosine rules for thesides
and the
five‐partformulas(I)simultaneously.
Figure5. Conversion of frame A to frame B using the
counterclockwiserotationmatrix.
2.3 Elementsubstitution
Otherformulasofthesinerules,thecosine rulesfor
thesidesandthefive‐partformulas canbeobtained
byusingtheelementsubstitution.InSection2.2,the
positionorderofelementsinthesphericaltriangleare
as shown in Figure 6(a). To replace the elements in
Figure 6(a) with the elements in Figure 6(b),
Equations5,6and7 canbereplacedtoEquations13,
14and15;Equations10,11and12canbereplacedto
Equations16,17and18respectively.
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Figure6.Positionorderofelements.
sin sin sin sinac
(13)
sin cos cos sin cos sin cosabcbc
(14)
cos sin sin cos cos cosabc bc
(15)
sin sin sin sinca
(16)
sin cos cos sin cos sin coscbaba
(17)
cos sin sin cos cos coscba ba
(18)
Equations13and16arethesinerules;Equations
14and17arethefive‐partformulas(I);Equations15
and18arethecosinerulesforthesides.
Similarly, to replace the elements in Figure 6(a)
withtheelementsinFigure6(c),Equations5,6and7
can
bereplacedtoEquations19,20and21;Equations
10,11and12canbereplacedtoEquations22,23and
24respectively.
sin sin sin sincb
(19)
sin cos cos sin cos sin coscabab
(20)
cos sin sin cos cos coscab ab
(21)
sin sin sin sinbc
(22)
sin cos cos sin cos sin cosbacac
(23)
cos sin sin cos cos cosbac ac
(24)
Equations19and22arethesinerules;Equations
20and23arethefive‐partformulas(I);Equations21
and24arethecosinerulesforthesides.
2.4 Derivationofothersphericaltriangleformulas
The other spherical triangle formulas can be easily
derived from the sine rules,
the cosine rules for the
sides,andthefive‐partformulas(I).Forexample,the
four‐partformulascanbederivedfromthesinerules
andthefive‐partformulas(I)asfollows:Substituting
Equation5into6canyieldEquation 25;substituting
Equation 10 into 11 can yield Equation 26;
substituting Equation 13 into 14 can yield Equation
27; substituting Equation 16 into 17 can yield
Equation 28; substituting Equation 19 into 20 can
yield Equation 29; substituting Equation 22 into 23
canyieldEquation30.
cos cos cot sin cot sincac

(25)
cos cos cot sin cot sincbc

(26)
cos cos cot sin cot sinbcb

(27)
cos cos cot sin cot sinbab

(28)
cos cos cot sin cot sinaba

(29)
cos cos cot sin cot sinaca

(30)
Equations 25 to 30 are the complete four‐part
formulas.
3 METHODSCOMPARISON
When we use the geometric method to derive the
fundamentalformulas,theprocessiscomplicatedand
different.AsshowninFigure7,wehavetodrawan
auxiliary plane triangle
ADE
outside the unit
sphere, and then use the plane cosine rules and
Pythagoreanequations toderive thecosinerulesfor
thesidesofthespheretriangle
ABC
.However,if
we want to use the geometric method to derive the
sine rules, the derivation process is completely
different.Wemustdrawtwoauxiliaryplanetriangles
insidetheunitsphere,asshowninFigure8.
222
DE AD AE 2AD AE
cos
222
DE OD OE 2OD OEcos
a
222
OD OA AD
222
OE OA AE
2222
DE 2OA AD AE 2OD OEcos
a
2
2AD AE 2OA 2OD OE
cos cos
a
2
OA AD AE
OD OE OD OE


cos cos
a

cos cos cos sin sin cos
abcbc
EAD EOD
,
a
AD OA AE OA,
Figure7.Usinggeometricmethodtoderivethecosinerules
forthesides.
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PE
POE
OP
 sin sin
c
OA
PD
DE OB DF OC,
PD
PED
PE

sin sin
PE OP sin
c
PD PE OP
sin sin sin
c
PF
POF
OP
 sin sin
b
PF OP sin
b
PD
PFD
PF

sin sin
PD PF OP
sin sin sin
b
OP = OP
sin sin sin sin
cb
=
sin sin
sin sin
bc
Figure8.Usinggeometricmethodtoderivethesinerules.
In addition,as shown in Figure9and Figure 10,
using the vector method to derive the fundamental
formulasmaylettheprocessbecomesimpler,butthe
process is also different. In comparison, the three‐
dimensionalrotationmethod proposedin thispaper
can easily and simultaneously derive the sine rules,
the
cosine rules for the sides, and the five‐part
formulas (I). It is especially helpful to the teaching
process.

b
c


90 0
90 0
90










cos sin
cos cos
sin
c
bc
c


90
90
90









cos sin
cos cos
sin
b
cb
b

cos
bc a


cos cos sin sin cos
bc b c b c

cos cos cos sin sin cos
abcbc
Figure9.Usingvectormethodtoderivethecosinerulesfor
thesides.

a

b
c


  
sin
ab ac abac a
sin sin sin
cb a


 

ab ac abca abac


  
abca


 
sin sin sin
cb abc


 
sin sin sin
ac bac

sin sin
sin sin
ab
Figure10.Usingvectormethodtoderivethesinerules.
4 APPLICATIONOFSPHERICALTRIANGLE
FORMULAS
4.1 Problemofgreatcircletrack
Theproblemsinvolvingtherelationshipsbetweenthe
threesidesandthree anglesofthespherical triangle
can be solved by using the spherical triangle
formulas.Theproblemofgreatcircletrackis oneof
them. As shown in Figure
11, while adopting the
greatcircletrackformthepointFtothepointT,the
latitudeandthelongitudeofthetwopointsaregiven
tofindthegreatcircledistance(
FT
D )andtheinitial
courseangle(
C ).
Figure11.Problemofgreatcircletrack.
In this problem, the formula of the great circle
distancecanbeobtainedbyusingthecosinerulesfor
thesidesasshowninEquation31,andtheformulaof
theinitialcourseanglecanbeobtainedbyusingthe
four‐partformulasasshowninEquation32.
FT F T F T FT
cos sin sin cos cos cosD


(31)
FT
FT F FT
sin
tan
cos tan sin cos
C


(32)
4.2 Demonstratedexample
A ship leaves from New York (
φ
40 27.1
N,
73 49.4
W) to Cape Town
(
φ
33 53.3
S,
18 23.1
E). The latitude of the
departure point (
F
40.452
), the difference of
longitude(
FT
92.208

), and thelatitude ofthe
destination point (
T
33.888

) are given. The
captain wants to know the great circle distance and
theinitialcourse(Bowditch,1981).ByusingEquation
31, we can obtain the great circle distance which is
6762.7 nautical miles (
FT
112.712D ). By using
Equation32,wecanyieldtheinitialcoursewhichis
115.9 ( N115.942 EC
).
5 CONCLUSIONS
Inthispaper,weprovidethearchitecturediagramto
clarify therelationships of various spherical triangle
formulas and indicate that the sine rules and the
cosine rules for the sides are the fundamental
formulas to derive the other spherical triangle
formulas. In addition, we propose the three‐
dimensional
rotation method which can derive the
sinerules,thecosinerulesforthesides,andthefive‐
partformulas(I)simultaneously.Themethodiseasier
tounderstandandcanimprovetheteachingprocess.
Furthermore, in practical use, we also demonstrate
howtousethesphericaltriangleformulastosolvethe
spherical navigation problems, such as finding the
greatcircledistanceandtheinitialcourseangle.
558
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Company.
Smart W. M. (1977). Text‐Book on Spherical Astronomy,
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Green R. M. (1985). Spherical Astronomy, Cambridge
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