# 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.