WOLFRAM|DEMONSTRATIONS PROJECT

Rational Linear Combinations of Pure Geodetic Angles, Part 2

​
a
0
1
a
1
1
a
2
1
a
3
1
b
0
1
b
1
1
b
2
1
b
3
1
d
1
2
3
5
7
11
d
2
3
5
7
11
13
-1
tan
1+
2
+
3
+
6

=
π
4
-
1
4
-1
tan
4
21
+
1
4
-1
tan
12
2
13
+
1
4
-1
tan
12
3
5
+
1
4
-1
tan
4
6
19
A "pure geodetic" angle is an angle
α
such that any of the six squared trigonometric functions of
α
is rational or infinite. This Demonstration shows how an angle whose tangent is of the form
b
0
/
a
0
+
b
1
/
a
1
d
1
+
b
2
/
a
2
d
2
+
b
3
/
a
3
d
1
d
2
can be expressed as a rational linear combination of pure geodetic angles and an integral multiple of
π/4
, that is, it finds rational
q
0
,
q
1
,
q
2
, and
q
3
, such that
-1
tan

b
0

a
0
+
b
1

a
1
d
1
+
b
2

a
2
d
2
+
b
3

a
3
d
1
d
2

is a sum of a rational linear combination of
-1
tan
(
q
0
)
,
-1
tan

q
1
d
1

,
-1
tan

q
2
d
2

, and
-1
tan

q
3
d
1
d
2

plus an integer multiple of
π/4
.