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(iii) The third identity (spherical side cosine lemma):
sin
β
=cos
ψ
=
z
0
z
0
=
r
sin
β
=
r
cos
ψ
r
,
sin
β
=sin
ψ
=cos
φ
cos
φ
0
cos
λ
cos
λ
0
+cos
φ
cos
φ
0
sin
λ
sin
λ
0
+sin
φ
sin
φ
0
,
⇒
(3.50)
sin
β
=sin
ψ
=cos
φ
cos
φ
0
cos(
λ
−
λ
0
)+sin
φ
sin
φ
0
.
Box 3.6 (The forward problem of transforming spherical frames of reference. Input variables:
λ, φ, λ
0
,φ
0
. Output variables:
α,β
).
(i) The first and second identities:
cos
φ
sin(
λ
−
λ
0
)
tan
α
=
sin
φ
cos
φ
0
,
cos
φ
sin
φ
0
cos(
λ
−
λ
0
)
−
sin(
λ
−
λ
0
)
tan
α
=
sin
φ
0
cos(
λ − λ
0
)
−
tan
φ
cos
φ
0
.
(3.51)
Alternatives:
1
cos
β
cos
φ
sin(
λ − λ
0
)
,
sin
α
=
1
cos
β
(cos
φ
sin
φ
0
cos(
λ
cos
α
=
−
λ
0
)
−
sin
φ
cos
φ
0
)
.
(3.52)
(ii) The third identity:
sin
β
=sin
ψ
=cos
φ
cos
φ
0
cos(
λ
−
λ
0
)+sin
φ
sin
φ
0
.
(3.53)
2
Box 3.7 (Representation of a placement vector
x
∈
S
r
in both the equatorial frame of refer-
ence
{
e
1
,
e
2
,
e
3
|O}
and the meta-equatorial (oblique) frame of reference
{
e
1
0
,
e
2
0
,
e
3
0
|O}
).
e
1
0
=+
e
1
sin
φ
0
cos
λ
0
+
e
2
sin
φ
0
sin
λ
0
−
e
3
cos
φ
0
,
e
2
0
=
−
e
1
sin
λ
0
+
e
2
cos
λ
0
,
(3.54)
e
3
0
=+
e
1
cos
φ
0
cos
λ
0
+
e
2
cos
φ
0
sin
λ
0
+
e
3
sin
φ
0
,
x
=
e
1
x
+
e
2
y
+
e
3
z
=
e
1
0
x
0
+
e
2
0
y
0
+
e
3
0
z
0
=
=
e
1
(
x
0
sin
φ
0
cos
λ
0
− y
0
sin
λ
0
+
z
0
cos
φ
0
cos
λ
0
)+
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