# Angular Difference and Pythagorean Addition
Angular relation is one important descriptor system.
A lone ultrareal has natural inner state $u$. When orientation is needed, an
inner state may be presented as:
$$
a=ue^{i\alpha}.
$$
For two oriented presentations,
$$
a=ue^{i\alpha},\qquad b=ve^{i\beta},
$$
the relevant quantity is not either angle alone. It is their relative
difference:
$$
\Delta=\alpha-\beta.
$$
The angular descriptor is:
$$
d=\cos\Delta.
$$
The joined value is:
$$
|a+b|^2
=u^2+v^2+2uv\cos\Delta.
$$
This is the relation-aware sum:
$$
U\oplus_{\cos\Delta}V.
$$
## Aligned Difference
When:
$$
\Delta=0,
$$
then:
$$
\cos\Delta=1.
$$
So:
$$
U\oplus_1V=(u+v)^2.
$$
This is aligned addition.
## Orthogonal Difference
When:
$$
\Delta=\frac{\pi}{2},
$$
then:
$$
\cos\Delta=0.
$$
So:
$$
U\oplus_0V=u^2+v^2.
$$
This is the Pythagorean case. The relation term vanishes.
If:
$$
c^2=u^2+v^2,
$$
then:
$$
c=\sqrt{u^2+v^2}.
$$
The Pythagorean theorem is therefore recovered as zero angular relation. In
ordinary geometry that condition is called orthogonality.
## Opposed Difference
When:
$$
\Delta=\pi,
$$
then:
$$
\cos\Delta=-1.
$$
So:
$$
U\oplus_{-1}V=(u-v)^2.
$$
Opposition is a relative difference between positive inner magnitudes. It is not
a negative ultrareal.
## Law Of Cosines
The same expression recovers the law of cosines. For joined oriented inner
states:
$$
|u+ve^{i\Delta}|^2
=u^2+v^2+2uv\cos\Delta.
$$
For the side opposite an angle $C$, the conventional triangle formula is:
$$
c^2=u^2+v^2-2uv\cos C.
$$
The sign difference is an orientation convention. The content is the same:
the square-value depends on the relation between the parts.
## Angle Addition
Euler's rotation rule gives the standard angle-addition formulas:
$$
e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}.
$$
Using:
$$
e^{i\theta}=\cos\theta+i\sin\theta,
$$
and collecting real and imaginary parts gives:
$$
\cos(\alpha+\beta)
=\cos\alpha\cos\beta-\sin\alpha\sin\beta,
$$
and:
$$
\sin(\alpha+\beta)
=\sin\alpha\cos\beta+\cos\alpha\sin\beta.
$$
Thus trigonometry is compatible with the ultrareal reading: angles describe how
inner presentations differ before a positive square-value is evaluated.