# Rational Linear Combinations of Pure Geodetic Angles

Rational Linear Combinations of Pure Geodetic Angles

A "pure geodetic" angle is an angle with any of its six squared trigonometric functions rational (or infinite). This Demonstration shows how an angle whose tangent is of the form /+/ can be expressed as a rational linear combination of pure geodetic angles and an integral multiple of , that is, it finds rational and such that + is a sum of , where and a rational linear combination of and .

b

1

a

1

d

1

b

2

a

2

d

2

π/2

q

1

q

2

-1

tan

b

1

a

1

d

1

b

2

a

2

d

2

kπ/2

k∈{-1,0,1}

-1

tan

q

1

d

1

-1

tan

q

2

d

2