Relation between the side lengths and angles of a spherical triangle
Spherical triangle In spherical trigonometry , the half side formula relates the angles and lengths of the sides of spherical triangles , which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles .[1]
For a triangle
△
A
B
C
{\displaystyle \triangle ABC}
[2]
tan
1
2
a
=
−
cos
(
S
)
cos
(
S
−
A
)
cos
(
S
−
B
)
cos
(
S
−
C
)
{\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos(S)\,\cos(S-A)}{\cos(S-B)\,\cos(S-C)}}}\end{aligned}}}
where a, b, c are the angular lengths (measure of central angle , arc lengths normalized to a sphere of unit radius ) of the sides opposite angles A, B, C respectively, and
S
=
1
2
(
A
+
B
+
C
)
{\displaystyle S={\tfrac {1}{2}}(A+B+C)}
b
{\displaystyle b}
c
{\displaystyle c}
A
,
B
,
C
.
{\displaystyle A,B,C.}
The polar dual relationship for a spherical triangle is the half-angle formula ,
tan
1
2
A
=
sin
(
s
−
b
)
sin
(
s
−
c
)
sin
(
s
)
sin
(
s
−
a
)
{\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin(s)\,\sin(s-a)}}}\end{aligned}}}
where semiperimeter
s
=
1
2
(
a
+
b
+
c
)
{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
A
,
B
,
C
.
{\displaystyle A,B,C.}
Half-tangent variant [ edit ] The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If
t
a
=
tan
1
2
a
,
{\displaystyle t_{a}=\tan {\tfrac {1}{2}}a,}
t
b
=
tan
1
2
b
,
{\displaystyle t_{b}=\tan {\tfrac {1}{2}}b,}
t
c
=
tan
1
2
c
,
{\displaystyle t_{c}=\tan {\tfrac {1}{2}}c,}
t
A
=
tan
1
2
A
,
{\displaystyle t_{A}=\tan {\tfrac {1}{2}}A,}
t
B
=
tan
1
2
B
,
{\displaystyle t_{B}=\tan {\tfrac {1}{2}}B,}
t
C
=
tan
1
2
C
,
{\displaystyle t_{C}=\tan {\tfrac {1}{2}}C,}
t
a
2
=
(
t
B
t
C
+
t
C
t
A
+
t
A
t
B
−
1
)
(
−
t
B
t
C
+
t
C
t
A
+
t
A
t
B
+
1
)
(
t
B
t
C
−
t
C
t
A
+
t
A
t
B
+
1
)
(
t
B
t
C
+
t
C
t
A
−
t
A
t
B
+
1
)
.
{\displaystyle {\begin{aligned}t_{a}^{2}&={\frac {{\bigl (}t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}-1{\bigr )}{\bigl (}{-t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}+1}{\bigr )}}{{\bigl (}t_{B}t_{C}-t_{C}t_{A}+t_{A}t_{B}+1{\bigr )}{\bigl (}t_{B}t_{C}+t_{C}t_{A}-t_{A}t_{B}+1{\bigr )}}}.\end{aligned}}}
and the half-angle formula is equivalent to:
t
A
2
=
(
t
a
−
t
b
+
t
c
+
t
a
t
b
t
c
)
(
t
a
+
t
b
−
t
c
+
t
a
t
b
t
c
)
(
t
a
+
t
b
+
t
c
−
t
a
t
b
t
c
)
(
−
t
a
+
t
b
+
t
c
+
t
a
t
b
t
c
)
.
{\displaystyle {\begin{aligned}t_{A}^{2}&={\frac {{\bigl (}t_{a}-t_{b}+t_{c}+t_{a}t_{b}t_{c}{\bigr )}{\bigl (}t_{a}+t_{b}-t_{c}+t_{a}t_{b}t_{c}{\bigr )}}{{\bigl (}t_{a}+t_{b}+t_{c}-t_{a}t_{b}t_{c}{\bigr )}{\bigl (}{-t_{a}+t_{b}+t_{c}+t_{a}t_{b}t_{c}}{\bigr )}}}.\end{aligned}}}
^ Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics ISBN 9783540721222 [1] ^ Nelson, David (2008), The Penguin Dictionary of Mathematics ISBN 9780141920870
