Phase Angle Calculator – Cos φ, Sin φ, Tan φ Power Factor
Description
Calculate phase angle φ in degrees or radians and its sine, cosine, and tangent values. Ideal for AC circuit analysis, power factor calculation, motor performance evaluation, and electrical engineering education. Supports bidirectional conversion with high precision.
Supported Parameters & Relationships
| Parameter | Meaning | Mathematical Relation |
|---|---|---|
| φ (°) | Phase angle in degrees | Angle between voltage and current, in degrees |
| φ (Rad) | Phase angle in radians | φ_rad = φ_deg × π / 180 |
| Sin φ | Sine of φ | sin(φ) |
| Cos φ | Cosine of φ (Power Factor) | cos(φ), i.e., Power Factor PF = P/S |
| Tan φ | Tangent of φ | tan(φ) = sin(φ)/cos(φ) |
Key Conversion Formulas
- φ (rad) = φ (°) × π / 180
- φ (°) = φ (rad) × 180 / π
- sin(φ) — sine of the phase angle
- cos(φ) — cosine of the phase angle; also known as Power Factor (PF)
- tan(φ) = sin(φ) / cos(φ)
Example Calculations
Example 1: Given φ = 30°
- φ (Rad): 30 × π / 180 ≈ 0.5236 rad
- sin φ: sin(30°) ≈ 0.5
- cos φ: cos(30°) ≈ 0.866 (Power Factor)
- tan φ: tan(30°) ≈ 0.577
Example 2: Given cos φ = 0.8
- φ: arccos(0.8) ≈ 36.87°
- φ (Rad): 36.87 × π / 180 ≈ 0.6435 rad
- sin φ: sin(36.87°) ≈ 0.6
- tan φ: tan(36.87°) ≈ 0.75
Example 3: Given tan φ = 1.732
- φ: arctan(1.732) ≈ 60°
- cos φ: cos(60°) = 0.5 (Power Factor)
- sin φ: sin(60°) ≈ 0.866
Example 4: Given φ = 90°
- φ (Rad): 90 × π / 180 = 1.5708 rad
- sin φ: sin(90°) = 1.0
- cos φ: cos(90°) = 0.0
- tan φ: tan(90°) → undefined (infinite)
Use Cases
- Power system design and analysis — calculating reactive power and apparent power
- Motor and transformer performance evaluation — assessing efficiency and losses
- Power quality monitoring — detecting poor power factor and implementing correction
- Industrial energy efficiency optimization — reducing electricity costs via capacitors
- Electrical engineering exams and teaching — solving AC circuit problems
- Electronic circuit simulation and debugging — verifying phase relationships in filters and amplifiers
Frequently Asked Questions
What is the phase angle φ in an AC circuit?
The phase angle φ is the angular difference between the voltage and current waveforms in an AC circuit.
It indicates how much the current leads or lags the voltage.
- φ = 0°: Purely resistive load (e.g., heater)
- φ > 0°: Inductive load (e.g., motor, transformer)
- φ < 0°: Capacitive load (e.g., capacitor bank)
How is cos φ related to power factor?
cos φ is the power factor (PF) in AC circuits.
It represents the ratio of real power (P) to apparent power (S):
PF = P / S = cos(φ)
A higher cos φ means better efficiency and less wasted energy.
Why is a low power factor bad?
A low power factor increases current flow for the same amount of real power, leading to:
- Higher I²R losses in conductors
- Larger conductor sizes needed
- Increased transformer and generator loading
- Higher electricity bills (some utilities charge penalties)
Power factor correction using capacitors can improve it.
Can this tool calculate φ from sin φ or tan φ?
Yes! You can input any one value:
- If you know sin φ, the tool calculates φ = arcsin(sin φ)
- If you know tan φ, it calculates φ = arctan(tan φ)
- Then computes all other values including cos φ and φ in degrees/radians.
What happens when cos φ = 0?
When cos φ = 0, the phase angle φ = ±90°.
This means the circuit is purely reactive (either inductive or capacitive).
Real power (P) = 0, so no useful work is done.
All power is reactive (Q), which oscillates between source and load.
How do I interpret negative phase angles?
A negative phase angle means the current leads the voltage.
This occurs in capacitive loads, such as capacitor banks or certain types of electronic devices.
In contrast, inductive loads (like motors) have positive φ (current lags).
Is there a limit to the phase angle φ?
Yes. In practical AC systems, φ typically ranges from -90° to +90°.
- At ±90°, cos φ = 0 → no real power
- At 0°, cos φ = 1 → maximum efficiency
Most industrial systems aim for φ close to 0° (high power factor).
Reference Standards
- IEEE 1459: Definitions for the Measurement of Electric Power and Energy
- IEC 61000-4-30: Power Quality Measurement
- NEC (National Electrical Code) – Article 450: Transformers
- Textbooks: "Electric Circuits" by James W. Nilsson, "Power System Analysis" by John J. Grainger
