Systemic Docs

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

Name	Type	Description
ArcSin	floatarray	The arcsine of each input value, expressed in radians.

The Arcsine module computes the inverse sine of each value in a data series, returning the angle in radians whose sine equals that value. With its symmetric output centered on zero, arcsine is particularly powerful for analyzing oscillating indicators and extracting cycle phase information.

y = \arcsin(x) \quad \text{where} \quad \sin(y) = x

Mathematical Foundation

Domain and Range

flowchart LR
    subgraph Domain["Input Domain"]
        D["-1 ≤ x ≤ 1"]
    end
    
    subgraph Transform["ARCSIN"]
        F["arcsin(x)"]
    end
    
    subgraph Range["Output Range"]
        R["-π/2 to +π/2 radians<br/>(-90° to +90°)"]
    end
    
    D --> F --> R

Property	Value
Domain	$x \in [-1, 1]$
Range	$y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ radians (-90° to 90°)

Key Advantage: Unlike arccos, arcsine output is symmetric around zero. This makes it ideal for indicators where zero-crossings are meaningful.

Key Values Reference

Input $x$	Output (radians)	Output (degrees)	Oscillator Interpretation
1	1.571	90°	Maximum bullish extreme
0.5	0.524	30°	Moderately bullish
0	0	0°	Neutral / zero-crossing
-0.5	-0.524	-30°	Moderately bearish
-1	-1.571	-90°	Maximum bearish extreme

Financial Application: Oscillator Phase Extraction

Many technical oscillators produce values that oscillate between -1 and 1 (when normalized). Arcsine can extract the phase angle of these oscillations, revealing where you are in the cycle.

Cycle Phase Trading

flowchart TD
    subgraph Input["Normalized Oscillator"]
        OSC["RSI normalized to [-1, 1]<br/>or Stochastic"]
    end
    
    subgraph Transform["Phase Extraction"]
        ARCSIN["ARCSIN Module"]
    end
    
    subgraph Phase["Cycle Position"]
        P90["+90° = Peak"]
        P0["0° = Midpoint"]
        N90["-90° = Trough"]
    end
    
    OSC --> ARCSIN
    ARCSIN --> P90 & P0 & N90

Interpretation:

Phase near +90°: Oscillator at maximum, possible exhaustion
Phase near 0°: Oscillator crossing equilibrium, momentum shift
Phase near -90°: Oscillator at minimum, possible reversal

Creative Use Case: Momentum Angle Indicator

Create an "angular momentum" indicator that emphasizes changes near the extremes:

flowchart LR
    subgraph System["Angular Momentum System"]
        RSI["RSI (0-100)"] --> NORM["Normalize to [-1, 1]"]
        NORM --> ARCSIN["ARCSIN"]
        ARCSIN --> DEG["Convert to Degrees"]
        DEG --> ROC["Rate of Change"]
        ROC --> Signal["Angular Velocity"]
    end

Why This Works

The arcsine transformation:

Compresses the middle — Small moves near 50 RSI create small angle changes
Expands the extremes — Small moves near 0 or 100 RSI create large angle changes
Highlights exhaustion — Slowing angular velocity at extremes signals potential reversals

Example Interpretation

RSI	Normalized	Arcsin (degrees)	Meaning
95	0.9	64°	Near peak, slowing
50	0	0°	Neutral momentum
5	-0.9	-64°	Near trough

Variance Stabilization: The Arcsine Transformation

In statistics, the arcsine transformation is used to stabilize variance in proportion data. In finance, this applies to:

Win rate percentages — Transform to stabilize analysis
Allocation weights — Analyze portfolio proportion changes
Market share data — Compare proportional changes fairly

\text{Transformed} = \arcsin(\sqrt{p})

Where $p$ is a proportion between 0 and 1.

Handling Invalid Inputs

Arcsine is only defined for values between -1 and 1. If an input value falls outside $[-1, 1]$ , the output at that position is null — the math is undefined there, and the module never substitutes a fake number. To avoid null outputs from out-of-range data:

Pre-normalize data to ensure valid range
Use CLAMP module to bound inputs before transformation

Missing Data

When an input position is missing (null) — for example a data gap or a closed-market day — the output at that same position is null. The module never fills in a value for data it does not have, so every non-null output is real math on real data.

Note the distinction: a null input means no datapoint exists here, while an out-of-range input means a datapoint exists but arcsine is undefined for it. Both surface as null in the output.

In live execution the module returns exactly one datapoint — the newest position — which is null when that position is missing or out of range.

Converting Radians to Degrees

The output is in radians. To convert to degrees:

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

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

ArcSine

Inputs

Outputs