**6. Other Conjugate Fields**

Here
are some useful conjugate field operators: 2 arctan[+], arccos[*] , tanh[+],
tan[+].

**; for subtended angles.**

*2 arctan[+]*
If
a line segment of length L is held with midpoint at distance D away from an
observer, at right angles to the observer's line of sight, then it will subtend
an angle of

θ = 2
arctan ( L/(2D) ) .

Therefore
two lengths, which subtend angles of (respectively) θ

_{1}and θ_{2}, when joined together will subtend an angle of
θ

_{12}= θ_{1}(2 arctan)[+] θ_{2}**; for spherical right triangles.**

*arccos[*]*
If
a spherical right triangle has sides A, B and hypotenuse C, (as measured in
radians), then we have the equation:

cos(A)
* cos(B) = cos(C)

Ergo:

C =
A arccos[*] B

**; for relativistic velocity addition.**

*tanh[+]*
The
relativistic velocity addition law can be written with tanh[+]:

V

_{12}= ( V_{1}+ V_{2}) / ( 1 + V_{1}V_{2}/c^{2}) = c ( V_{1}/c tanh[+] V_{2}/c )**; for slope rotation.**

*tan[+]*
If
two lines have slopes x and y, and a third line is tilted from the horizontal
at an angle equal to the sum of the other two line’s tilts, then the third line
has this slope:

tan(
arctan(x) + arctan(y)) = ( x + y ) / ( 1 – x y )

Note
that ( x tan[+] y ) = i ( x/i
+ y/i ) / ( 1 + (x/i)(y/i) ) = i ( (x/i) tanh[+] (y/i))

Therefore
: tan[+] =
(i*tanh)[+]

And
likewise: tanh[+] =
(i*tan)[+]

Slope
rotation and relativistic velocity addition are imaginary conjugates of each
other!

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