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cos
α
=
-
cos
β
cos
γ
+
sin
β
sin
γ
cosh
a
k
,
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Doc 1
1.0000
1.0000
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Hyperbolic_law_of_cosines.html
cos
α
=
-
cos
β
cos
γ
+
sin
β
sin
γ
cosh
a
k
,
Doc 2
0.3333
0.3333
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Hyperbolic_triangle.html
cos
C
=
-
cos
A
cos
B
+
sin
A
sin
B
cosh
c
,
Doc 3
0.2667
1.3535
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Spherical_trigonometry.html
cos
A
=
-
cos
B
cos
C
+
sin
B
sin
C
cos
a
,
cos
B
=
-
cos
C
cos
A
+
sin
C
sin
A
cos
b
,
cos
C
=
-
cos
A
cos
B
+
sin
A
sin
B
cos
c
.
cos
a
=
cos
b
cos
c
+
sin
b
sin
c
cos
A
,
cos
b
=
cos
c
cos
a
+
sin
c
sin
a
cos
B
,
cos
c
=
cos
a
cos
b
+
sin
a
sin
b
cos
C
,
cos
a
=
(
cos
a
cos
c
+
sin
a
sin
c
cos
B
)
cos
c
+
sin
b
sin
c
cos
A
cos
a
=
cos
b
cos
c
+
sin
b
sin
c
cos
A
.
cos
a
sin
2
c
=
sin
a
cos
c
sin
c
cos
B
+
sin
b
sin
c
cos
A
Doc 4
0.2667
0.5624
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Law_of_cosines.html
cos
A
=
-
cos
B
cos
C
+
sin
B
sin
C
cosh
a
.
sin
(
α
+
γ
)
=
sin
α
cos
γ
+
sin
γ
cos
α
cos
2
γ
+
sin
2
γ
=
1.
c
⋅
(
sin
α
cos
γ
+
sin
γ
cos
α
)
=
b
sin
γ
,
c
=
a
cos
β
+
b
cos
α
.
Doc 5
0.1806
0.2528
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Spherical_law_of_cosines.html
cos
(
A
)
=
-
cos
(
B
)
cos
(
C
)
+
sin
(
B
)
sin
(
C
)
cos
(
a
)
cos
(
c
)
=
cos
(
a
)
cos
(
b
)
+
sin
(
a
)
sin
(
b
)
cos
(
C
)
.
Doc 6
0.1511
1.5509
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Proofs_of_trigonometric_identities.html
cos
(
α
-
β
)
=
cos
α
cos
β
+
sin
α
sin
β
cos
(
α
-
β
)
=
cos
α
cos
β
+
sin
α
sin
β
cos
(
α
-
β
)
=
cos
α
cos
-
β
-
sin
α
sin
-
β
cos
(
α
+
β
)
=
cos
α
cos
β
-
sin
α
sin
β
cos
(
α
+
β
)
=
cos
α
cos
β
-
sin
α
sin
β
sin
(
α
-
β
)
=
sin
α
cos
-
β
+
cos
α
sin
-
β
e
i
(
α
+
β
)
=
(
cos
α
cos
β
-
sin
α
sin
β
)
+
i
(
sin
α
cos
β
+
sin
β
cos
α
)
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
sin
(
α
+
β
)
=
sin
α
cos
β
+
sin
β
cos
α
cos
(
α
+
β
)
+
cos
(
α
-
β
)
=
cos
α
cos
β
-
sin
α
sin
β
+
cos
α
cos
β
+
sin
α
sin
β
=
2
cos
α
cos
β
cos
(
α
+
β
)
-
cos
(
α
-
β
)
=
cos
α
cos
β
-
sin
α
sin
β
-
cos
α
cos
β
-
sin
α
sin
β
=
-
2
sin
α
sin
β
sin
(
α
-
β
)
=
sin
α
cos
β
-
cos
α
sin
β
sin
(
α
-
β
)
=
sin
α
cos
β
-
cos
α
sin
β
cot
(
α
+
β
)
=
cos
(
α
+
β
)
sin
(
α
+
β
)
=
cos
α
cos
β
-
sin
α
sin
β
sin
α
cos
β
+
cos
α
sin
β
tan
(
α
+
β
)
=
sin
(
α
+
β
)
cos
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
cos
α
cos
β
-
sin
α
sin
β
sin
(
α
+
β
)
-
sin
(
α
-
β
)
=
sin
α
cos
β
+
cos
α
sin
β
-
sin
α
cos
β
+
cos
α
sin
β
=
2
cos
α
sin
β
cos
(
α
+
β
)
=
O
B
=
O
A
-
B
A
=
O
A
-
R
Q
=
cos
α
cos
β
-
sin
α
sin
β
Doc 7
0.1488
0.7948
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Solution_of_triangles.html
a
=
arccos
(
cos
α
+
cos
β
cos
γ
sin
β
sin
γ
)
a
=
arccos
(
cos
α
+
cos
β
cos
γ
sin
β
sin
γ
)
,
c
=
arccos
(
cos
a
cos
b
+
sin
a
sin
b
cos
γ
)
γ
=
arccos
(
sin
α
sin
β
cos
c
-
cos
α
cos
β
)
,
c
=
arccos
(
cos
γ
+
cos
α
cos
β
sin
α
sin
β
)
.
α
=
180
∘
-
β
-
γ
b
=
arccos
(
cos
β
+
cos
γ
cos
α
sin
γ
sin
α
)
b
=
arccos
(
cos
β
+
cos
γ
cos
α
sin
γ
sin
α
)
,
sin
γ
=
c
b
sin
β
.
Doc 8
0.1300
0.2734
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Ptolemy's_theorem.html
sin
α
sin
β
cos
β
cos
γ
+
sin
α
cos
2
β
sin
γ
+
cos
α
sin
2
β
cos
γ
+
cos
α
sin
β
cos
β
sin
γ
sin
(
α
+
β
)
sin
(
β
+
γ
)
=
sin
α
sin
γ
+
sin
β
sin
(
α
+
β
+
γ
)
cos
(
x
+
y
)
=
cos
x
cos
y
-
sin
x
sin
y
Doc 9
0.1162
0.1738
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Celestial_coordinate_system.html
sin
δ
=
sin
β
cos
ε
+
cos
β
sin
ε
sin
λ
tan
α
=
sin
λ
cos
ε
-
tan
β
sin
ε
cos
λ
;
{
cos
δ
sin
α
=
cos
β
sin
λ
cos
ε
-
sin
β
sin
ε
;
cos
δ
cos
α
=
cos
β
cos
λ
.
Doc 10
0.1156
0.4325
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/List_of_trigonometric_identities.html
cos
(
α
+
β
)
=
cos
α
cos
β
-
sin
α
sin
β
cos
(
α
±
β
)
=
cos
α
cos
β
∓
sin
α
sin
β
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
sin
(
α
±
β
)
=
sin
α
cos
β
±
cos
α
sin
β
tan
(
α
+
β
2
)
=
sin
α
+
sin
β
cos
α
+
cos
β
=
-
cos
α
-
cos
β
sin
α
-
sin
β
Doc 11
0.1156
0.1956
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Machin-like_formula.html
cos
(
α
+
β
)
=
cos
α
cos
β
-
sin
α
sin
β
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
Doc 12
0.1098
0.1098
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Differential_geometry_of_surfaces.html
cos
c
=
cos
a
cos
b
+
sin
a
sin
b
cos
γ
.
Doc 13
0.1071
0.1071
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Bravais_lattice.html
a
b
c
1
-
cos
2
α
-
cos
2
β
-
cos
2
γ
+
2
cos
α
cos
β
cos
γ
Doc 14
0.1070
0.1070
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Solar_zenith_angle.html
sin
α
s
=
cos
h
cos
δ
cos
φ
+
sin
δ
sin
φ
Doc 15
0.1032
0.3694
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Universal_joint.html
𝐱
^
2
=
[
-
cos
β
sin
γ
2
,
cos
γ
2
,
sin
β
sin
γ
2
]
ω
2
=
ω
1
cos
β
1
-
sin
2
β
cos
2
γ
1
γ
2
=
atan2
(
sin
γ
1
,
cos
β
cos
γ
1
)
a
2
=
a
1
cos
β
1
-
sin
2
β
cos
2
γ
1
-
ω
1
2
cos
β
sin
2
β
sin
2
γ
1
(
1
-
sin
2
β
cos
2
γ
1
)
2
tan
γ
1
=
cos
β
tan
γ
2
Doc 16
0.1020
0.1020
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Catenoid.html
y
(
u
,
v
)
=
-
cos
θ
sinh
v
cos
u
+
sin
θ
cosh
v
sin
u
Doc 17
0.0996
0.1584
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Prosthaphaeresis.html
cos
a
=
cos
b
cos
c
+
sin
b
sin
c
cos
α
cos
a
-
cos
b
=
-
2
sin
(
a
+
b
2
)
sin
(
a
-
b
2
)
Doc 18
0.0978
0.2666
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Trigonometric_functions.html
cos
(
x
-
y
)
=
cos
x
cos
y
+
sin
x
sin
y
cos
(
x
-
y
)
=
cos
x
cos
y
+
sin
x
sin
y
.
cos
(
x
+
y
)
=
cos
x
cos
y
-
sin
x
sin
y
,
Doc 19
0.0924
0.2131
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Wetting.html
γ
α
θ
cos
α
+
γ
θ
β
cos
β
+
γ
α
β
=
0
γ
α
θ
+
γ
θ
β
cos
θ
+
γ
α
β
cos
α
=
0
γ
α
θ
cos
θ
+
γ
θ
β
+
γ
α
β
cos
β
=
0
Doc 20
0.0924
0.2131
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Ideal_surface.html
γ
α
θ
cos
α
+
γ
θ
β
cos
β
+
γ
α
β
=
0
γ
α
θ
+
γ
θ
β
cos
θ
+
γ
α
β
cos
α
=
0
γ
α
θ
cos
θ
+
γ
θ
β
+
γ
α
β
cos
β
=
0
Doc 21
0.0907
0.1813
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Triangle.html
cos
2
α
+
cos
2
β
+
cos
2
γ
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
=
1
,
r
R
=
4
T
2
s
a
b
c
=
cos
α
+
cos
β
+
cos
γ
-
1
;
Doc 22
0.0898
0.0898
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Spherical_harmonics.html
cos
(
θ
′
-
θ
)
=
cos
θ
′
cos
θ
+
sin
θ
sin
θ
′
Doc 23
0.0881
0.0881
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Orthonormality.html
cos
θ
1
cos
θ
2
+
sin
θ
1
sin
θ
2
=
0
Doc 24
0.0849
0.2780
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Plus-minus_sign.html
cos
(
A
+
B
)
=
cos
(
A
)
cos
(
B
)
+
sin
(
A
)
sin
(
B
)
cos
(
A
-
B
)
=
cos
(
A
)
cos
(
B
)
+
sin
(
A
)
sin
(
B
)
cos
(
A
-
B
)
=
cos
(
A
)
cos
(
B
)
-
sin
(
A
)
sin
(
B
)
cos
(
A
±
B
)
=
cos
(
A
)
cos
(
B
)
∓
sin
(
A
)
sin
(
B
)
Doc 25
0.0849
0.0849
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Rifleman's_rule.html
cos
(
θ
-
α
)
=
cos
(
θ
)
cos
(
α
)
+
sin
(
θ
)
sin
(
α
)
Doc 26
0.0804
0.1505
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Pythagorean_theorem.html
cos
(
c
R
)
=
cos
(
a
R
)
cos
(
b
R
)
+
sin
(
a
R
)
sin
(
b
R
)
cos
γ
.
cos
2
θ
+
sin
2
θ
=
a
2
+
b
2
c
2
=
1
,
Doc 27
0.0793
0.0793
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Tetrahedron.html
V
=
a
b
c
6
1
+
2
cos
α
cos
β
cos
γ
-
cos
2
α
-
cos
2
β
-
cos
2
γ
,
Doc 28
0.0765
0.0765
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Rigid_rotor.html
(
ω
x
ω
y
ω
z
)
=
(
-
sin
β
cos
γ
sin
γ
0
sin
β
sin
γ
cos
γ
0
cos
β
0
1
)
(
α
˙
β
˙
γ
˙
)
.
Doc 29
0.0756
0.0756
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Solar_irradiance.html
cos
(
c
)
=
cos
(
a
)
cos
(
b
)
+
sin
(
a
)
sin
(
b
)
cos
(
C
)
Doc 30
0.0745
0.0745
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Spherical_multipole_moments.html
cos
γ
=
cos
θ
cos
θ
′
+
sin
θ
sin
θ
′
cos
(
ϕ
-
ϕ
′
)
Doc 31
0.0743
0.0743
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Geodesics_on_an_ellipsoid.html
tan
α
1
=
cos
β
2
sin
ω
12
cos
β
1
sin
β
2
-
sin
β
1
cos
β
2
cos
ω
12
,
Doc 32
0.0727
0.0727
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Brahmagupta's_formula.html
cos
C
=
-
cos
A
Doc 33
0.0721
0.0721
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Haversine_formula.html
cos
(
c
)
=
cos
(
a
)
cos
(
b
)
+
sin
(
a
)
sin
(
b
)
cos
(
C
)
.
Doc 34
0.0714
0.0714
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Green's_function_for_the_three-variable_Laplace_equation.html
cos
γ
=
cos
θ
cos
θ
′
+
sin
θ
sin
θ
′
cos
(
φ
-
φ
′
)
.
Doc 35
0.0698
0.0698
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Rayleigh_sky_model.html
cos
γ
=
sin
θ
s
sin
θ
cos
ψ
+
cos
θ
s
cos
θ
Doc 36
0.0679
0.0679
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Sundial.html
sin
G
=
cos
L
cos
D
cos
R
-
sin
L
sin
R
=
-
cos
L
cos
D
sin
I
+
sin
L
cos
I
Doc 37
0.0667
0.0667
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Riemannian_connection_on_a_surface.html
d
α
=
β
∧
γ
,
d
β
=
γ
∧
α
,
d
γ
=
α
∧
β
.
Doc 38
0.0662
0.0662
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Classical_central-force_problem.html
s
y
m
b
o
l
φ
^
⋅
𝐫
^
=
-
sin
φ
cos
φ
+
cos
φ
sin
φ
=
0
Doc 39
0.0655
0.0655
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Laplace's_equation.html
cos
Θ
=
cos
φ
cos
φ
′
+
sin
φ
sin
φ
′
cos
(
θ
-
θ
′
)
.
Doc 40
0.0638
0.0638
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Tridiagonal_matrix.html
cos
(
x
)
=
-
cos
(
π
-
x
)
Doc 41
0.0622
0.0622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Abū_al-Wafā'_Būzjānī.html
sin
(
α
±
β
)
=
sin
α
cos
β
±
cos
α
sin
β
Doc 42
0.0622
0.0622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Trigonometry.html
cos
(
A
±
B
)
=
cos
A
cos
B
∓
sin
A
sin
B
Doc 43
0.0622
0.0622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Clausen_function.html
cos
(
x
-
y
)
=
cos
x
cos
y
-
sin
x
sin
y
Doc 44
0.0622
0.0622
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/History_of_trigonometry.html
sin
(
α
±
β
)
=
sin
α
cos
β
±
cos
α
sin
β
Doc 45
0.0621
0.0621
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Pauli_matrices.html
cos
c
=
cos
a
cos
b
-
n
^
⋅
m
^
sin
a
sin
b
,
Doc 46
0.0619
0.0619
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Spectral_theory_of_ordinary_differential_equations.html
cos
β
f
(
x
)
-
sin
β
f
′
(
x
)
=
0
Doc 47
0.0619
0.0619
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Phase_detector.html
α
-
β
≈
sin
(
α
-
β
)
=
sin
α
cos
β
-
sin
β
cos
α
Doc 48
0.0615
0.0615
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Pentagon.html
=
cos
72
cos
18
-
sin
72
sin
18
Doc 49
0.0588
0.0588
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Cayley–Hamilton_theorem.html
e
i
(
θ
/
2
)
(
n
^
⋅
σ
)
=
I
2
cos
θ
/
2
+
i
(
n
^
⋅
σ
)
sin
θ
/
2
,
Doc 50
0.0542
0.0542
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Vector_algebra_relations.html
𝐀
=
∥
𝐀
∥
(
cos
α
𝐢
^
+
cos
β
𝐣
^
+
cos
γ
𝐤
^
)
,
Doc 51
0.0542
0.0542
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Rotation_group_SO(3).html
cos
c
′
=
cos
a
′
cos
b
′
-
u
^
⋅
v
^
sin
a
′
sin
b
′
,
Doc 52
0.0541
0.0541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Trigonometric_tables.html
cos
(
x
±
y
)
=
cos
(
x
)
cos
(
y
)
∓
sin
(
x
)
sin
(
y
)