Spread compared to angle Rational trigonometry

the spread of 2 lines can measured in 4 equivalent positions.

unlike angle, can define relationship between rays emanating point, circular measure parametrization, , pair of lines can considered 4 pairs of rays, forming 4 angles, spread more fundamental in rational trigonometry, describing 2 lines single measure of rational function (see above). being equivalent square of sine of corresponding angle θ (and haversine of chord-based double-angle Δ = 2θ), spread of both angle , supplementary angle equal.

spread not proportional, however, separation between lines angle be; spreads of 0, 1/4, 1/2, 3/4, , 1 corresponding unevenly spaced angles 0°, 30°, 45°, 60° , 90°.

instead, (recalling supplementary property) 2 equal, co-terminal spreads determine third spread, value solution of triple spread formula triangle (or 3 concurrent lines) spreads of s, s , r:

(
2
s
+
r

)

2





=
2

(
2

s

2


+

r

2


)

+
4

s

2


r




4

s

2


+
4
s
r
+

r

2





=
4

s

2


+
2

r

2


+
4

s

2


r






{\displaystyle {\begin{aligned}(2s+r)^{2}&=2\left(2s^{2}+r^{2}\right)+4s^{2}r\\4s^{2}+4sr+r^{2}&=4s^{2}+2r^{2}+4s^{2}r\end{aligned}}}

giving quadratic polynomial (in s):

r

2


+
4

s

2


r
−
4
s
r



=
0





r

2


−
4
s
(
1
−
s
)
r



=
0






{\displaystyle {\begin{aligned}r^{2}+4s^{2}r-4sr&=0\\r^{2}-4s(1-s)r&=0\end{aligned}}}

and solutions

r
=
0

(

trivial

)


or


r
=
4
s
(
1
−
s
)


{\displaystyle r=0\quad ({\text{trivial}})\qquad {\text{or}}\qquad r=4s(1-s)}

this equivalent trigonometric identity :

sin

2


⁡
(
2
θ
)
=
4

sin

2


⁡
θ

(
1
−

sin

2


⁡
θ
)



{\displaystyle \sin ^{2}(2\theta )=4\sin ^{2}\theta \left(1-\sin ^{2}\theta \right)}

of angles θ, θ , 180° − 2θ of triangle, using

s

2


(
s
)
=

s

2



(

sin

2


⁡
θ
)

=

sin

2


⁡
(
2
θ
)
=
r
(
s
)


{\displaystyle s_{2}(s)=s_{2}\left(\sin ^{2}\theta \right)=\sin ^{2}(2\theta )=r(s)}

to denote second spread polynomial in s.

tripling spreads likewise involves triangle (or 3 concurrent lines) 1 spread of r (the previous solution), 1 spread of s , obtaining third spread polynomial, t in s. turns out be:

s

3


(
s
)
=
s
(
3
−
4
s

)

2


=
t
(
s
)


{\displaystyle s_{3}(s)=s(3-4s)^{2}=t(s)}

further multiples of basic spread of lines can generated continuing process using triple spread formula.

every multiple of spread rational rational, converse not apply. example, half-angle formula, 2 lines meeting @ 15° (or 165°) angle have spread of:

hav
⁡

(

30

∘


)

=

sin

2


⁡

(



30

∘


2


)

=



1
−
cos
⁡

30

∘



2


=



1
−



3

2



2


=



2
−


3



4


≈
0.0667.


{\displaystyle \operatorname {hav} \left(30^{\circ }\right)=\sin ^{2}\left({\frac {30^{\circ }}{2}}\right)={\frac {1-\cos 30^{\circ }}{2}}={\frac {1-{\frac {\sqrt {3}}{2}}}{2}}={\frac {2-{\sqrt {3}}}{4}}\approx 0.0667.}

and exists algebraic extension of rational numbers.