After studying the previous model of a system of coordinates, I realized that it was unnecessary to complicate it by switching the axis. Instead, let us examine the coordinate system without switching the axis, while keeping the circles of the Fibonacci spiral stacked on top of each other. We obtain the following.
Here, sinsx and cossx represent spiral coordinates (no, it was not a typo). Equivalent to how hyperbolic trig functions are named, the “s” in sinsx and cossx stand for “spiral”. My thought process before coming to this graph was to use rectangular coordinates, since this graph resembles a unit circle. That way we would get rid of units since they do not apply to the Fibonacci spiral, and the graph would be the same. However, we would not get trigonometric functions if we used a rectangular coordinate system. So I scratched that out of my notes. Looking back at the graph above, I realized that this is definitely not the graph of a unit circle, or polar coordinates. This is a whole different graph. Each quarter term here will be called a dimension, and they are represented by the following angles:
Dimension 1: 0 degrees to 90 degrees
Dimension 2: 90 degrees to 180 degrees
Dimension 3: 180 degrees to 270 degrees
Dimension 4: 270 degrees to 360 degrees
Dimension 5: 360 degrees to 450 degrees
Dimension 6: 450 degrees to 540 degrees
Dimension 7: 540 degrees to 630 degrees
Dimension 8: 630 degrees to 720 degrees…
And so on. Angles that repeat themselves at the limits of each dimension can be portrayed geometrically by the spiral itself, at the point where the previous circle is tangent to the following circle. In the picture, the black dots show where the circles are tangent to each other at 90 degrees, 180 degrees, 270 degrees and so on.
One would ask, what is the difference between spiral coordinates and polar coordinates since they look similar.
1- The axis change. Instead of “r” and “theta” we have “sinsx” and “cossx”.
2- The angles do not recur. For example, 20 degrees is not equivalent to 380 degrees.
Meanwhile, there are fundamental differences between spiral coordinates and the unit circle as well:
1- We are not using a unit, lengths change according to the Fibonacci sequence.
2- Angles do not recur. Every subsequent quarter circle is bigger.
3- Quadrants “take turns” every fourth term.
In pursuit of the relation between spiral trigonometric functions (or “Trigonacci functions”) and regular trigonometric functions, quarter terms were looked into in more depth.
Placing these quarter circles onto spiral coordinates, sinsx goes in place of Xn and cossx in place of the horizontal side of the triangle. So in the formula 
, we replace the unproven nth term formula with Binet’s formula (see previous post for the proof that the two are equivalent) and substitute Xn with sinsx. Now, we arrive to the formulas for the first spiral trigonometric functions:
, we replace the unproven nth term formula with Binet’s formula (see previous post for the proof that the two are equivalent) and substitute Xn with sinsx. Now, we arrive to the formulas for the first spiral trigonometric functions: 
Where x is the angle and n is the number of the term (i.e. the dimension).
Now that we have the formulas of Trigonacci, let’s calculate some well known angles:
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