Systemic Docs

Name	Type	Description
Data	floatarray	The input data series for the arctangent calculation.

Name	Type	Description
ArcTan	floatarray	The arctangent of each input value, expressed in radians.

The Arctangent (arctan) module computes the inverse tangent of each value in a data series, returning the angle in radians whose tangent equals that value.

y = \arctan(x) \quad \text{where} \quad \tan(y) = x

Mathematical Foundation

Domain and Range

Property	Value
Domain	$x \in (-\infty, +\infty)$ — all real numbers
Range	$y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ radians (-90° to 90°)

Key Advantage: Unlike arcsine and arccosine, arctangent accepts any real number as input. This makes it ideal for processing unbounded data series without worrying about invalid inputs.

Properties

$\arctan(0) = 0$
$\arctan(1) = \frac{\pi}{4}$ (45°)
$\arctan(-1) = -\frac{\pi}{4}$ (-45°)
$\tan(\arctan(x)) = x$ for all $x$
As $x \to +\infty$ , $\arctan(x) \to \frac{\pi}{2}$
As $x \to -\infty$ , $\arctan(x) \to -\frac{\pi}{2}$

Key Values

Input $x$	Output (radians)	Output (degrees)
10	1.471	84.3°
1	0.785	45°
0.5	0.464	26.6°
0	0	0°
-0.5	-0.464	-26.6°
-1	-0.785	-45°
-10	-1.471	-84.3°

Behavior in Systemic

The Arctangent module applies the arctan function element-wise to every value in the input series.

Handling Any Input

Unlike arcsine and arccosine, arctangent is defined for all real numbers. Any numeric input produces a valid angle — even infinite values map cleanly to $\pm\frac{\pi}{2}$ . This makes arctangent particularly robust for processing raw, unbounded data.

Missing Data

If a value is missing (null) at some position in the input series — a data gap or a closed-market day — the output is null at that same position. The module never substitutes a fabricated number for missing data; every non-missing input still produces its real angle.

Because arctangent has no invalid inputs, a null output always means the input was missing at that position — never a math error.

In live execution, the module returns exactly one datapoint: the newest value, which is null when the newest input is missing.

Creative Applications

Data Normalization (Squashing)

Arctangent is an excellent squashing function that compresses any unbounded range into a bounded interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ :

Large positive values → approach $+\frac{\pi}{2}$
Large negative values → approach $-\frac{\pi}{2}$
Values near zero → approximately linear response

This property is invaluable for:

Outlier compression — reducing the impact of extreme values
Signal normalization — converting unbounded indicators into bounded ranges
Rate-of-change smoothing — compressing large price moves

Momentum Analysis

Apply arctangent to rate-of-change or momentum indicators to create a bounded momentum oscillator. Extreme momentum readings (both positive and negative) get compressed, making the signal easier to interpret and compare across different time periods or assets.

flowchart LR
    Price[Price Data] --> ROC[Rate of Change]
    ROC --> ARCTAN((Arctan))
    ARCTAN --> Bounded["Bounded Momentum<br/>(-π/2 to +π/2)"]

Converting Radians to Degrees

The output is in radians. To convert to degrees, multiply by $\frac{180}{\pi}$ :

\text{degrees} = \text{radians} \times \frac{180}{\pi} \approx \text{radians} \times 57.2958

Use a Multiply module with a constant of approximately 57.2958 to convert.

ArcTangent

Inputs

Outputs