φ(rad) = φ(°) × π / 180
φ(°) = φ(rad) × 180 / π
tan(φ) = sin(φ) / cos(φ)
sin²(φ) + cos²(φ) = 1
φ = arcsin(sin(φ))
φ = arccos(cos(φ))
φ = arctan(tan(φ))
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.
| 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) |
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)
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
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
φ: arctan(1.732) ≈ 60°
cos φ: cos(60°) = 0.5 (Power Factor)
sin φ: sin(60°) ≈ 0.866
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)
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
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
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)
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.
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.
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.
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.
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).
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).
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