The Angle Converter (Sin/Cos/Tan) for Electrical converts AC phase angles between degrees and radians while computing sine, cosine, and tangent values. Used for power factor calculations, impedance analysis, phasor representation, and AC circuit design in electrical engineering.
0.785398
rad
0.707107
0.707107
1
0.7071
0.785398
rad
0.707107
0.707107
1
0.7071
The calculator for trigonometric angle conversion in electrical applications converts phase angles between degrees and radians and computes the corresponding sine, cosine, and tangent values simultaneously. In AC electrical engineering, angles are not abstract — they define the phase relationship between voltage and current that determines power factor, reactive power, and circuit impedance.
In AC circuits, the phase angle φ between voltage and current waveforms determines how electrical power distributes between useful work and reactive oscillation:
At φ = 0° (unity power factor): PF = cos(0°) = 1.0; all power is active. At φ = 30°: PF = cos(30°) = 0.866; 13.4% of apparent power is reactive. At φ = 45°: PF = 0.707; equal active and reactive components. Use this online calculator to find all trig values for any phase angle. The active/reactive/apparent power calculator applies these relationships to complete power triangle analysis.
AC circuit impedance is a complex quantity with magnitude Z and phase angle φ:
Converting between polar (Z, φ) and rectangular (R, X) representations requires the same sin and cos operations this calculator provides. The delta-Y power converter and electrical conversion tools provide complementary AC circuit analysis calculators.
Electrical engineering convention uses degrees for phase angles in power systems (a 30° lagging power factor is more intuitive than 0.524 rad lagging). However, mathematical analysis of AC circuits, Fourier transforms, and control systems uses radians because angular frequency ω = 2πf is in rad/s, and the complex exponential e^(jωt) requires radians. Engineers must switch between these conventions fluidly, making this converter a practical everyday tool.
In balanced three-phase systems, the three voltage and current phasors are separated by exactly 120° (2π/3 radians). The symmetry of three-phase power — that the sum of three balanced phasors is always zero — depends directly on the trigonometric identity: sin(θ) + sin(θ + 120°) + sin(θ + 240°) = 0. Understanding the phase angles between phasors is fundamental to three-phase power analysis, fault calculation, and protection relay coordination.
The conversion from degrees to radians uses the fundamental relationship: θ_rad = θ_deg × π / 180. This stems from the definition that a full circle of 360° equals 2π radians, so each degree equals π/180 radians.
Once the radian value is obtained, JavaScript's built-in Math.sin(), Math.cos(), and Math.tan() functions compute the trigonometric values. These functions use the IEEE 754 double-precision floating-point standard, giving approximately 15 significant digits of accuracy — more than sufficient for any electrical engineering application.
The power factor output is computed as |cos θ| — the absolute value — because power factor is always reported as a positive number between 0 and 1, with the lagging/leading qualifier stated separately. A phase angle of +36.87° (lagging) and −36.87° (leading) both correspond to a power factor magnitude of 0.8.
The tangent is guarded against division-by-zero: when |cos θ| < 10⁻¹⁰ (essentially 90° or 270°), the result is reported as Infinity, which correctly represents the undefined nature of tan(90°).
Angle Rad: Use this value when inputting phase angles into engineering formulas that require radians (most programming languages, MATLAB, Python NumPy all use radians by default).
sin θ: Represents the reactive (imaginary) component of a unit phasor. Multiply by apparent power S to get reactive power Q.
cos θ: Represents the active (real) component of a unit phasor. This is the power factor of a load with this phase angle. Values close to 1.0 indicate efficient power utilization; values below 0.85 often trigger utility penalties in commercial/industrial settings.
tan θ: The ratio Q/P. Useful when calculating capacitor size for power factor correction: Q_cap = P × (tan θ_old − tan θ_new).
Power Factor: Directly read off for load characterization. A value of 1.0 is ideal (purely resistive); 0.0 means purely reactive load consuming no real power.
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A classic motor load angle. cos(36.87°) = 0.8 confirms 0.8 power factor. sin(36.87°) = 0.6 means Q/S = 0.6, so for every 1 kVA of apparent power, 0.8 kW is real and 0.6 kVAR is reactive.
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At 45°, sin and cos are equal (both √2/2 ≈ 0.7071). Real and reactive power are equal. Power factor = 0.707. This is often used as a design boundary in conservative power system planning.
Engineering conventions, instrumentation, and industry standards (IEEE, IEC) traditionally express phase angles in degrees because they are more intuitive for humans reading phasor diagrams and protection relay settings. Radians are mathematically cleaner but less practical for field work. This calculator accepts degrees as input and internally converts to radians for computation.
Power factor (PF) is the ratio of real power (W) to apparent power (VA): PF = P / S = cos θ, where θ is the phase angle between the voltage and current waveforms. A PF of 1.0 means voltage and current are perfectly in phase (resistive load). Lower PF means more reactive power, leading to higher current for the same useful power output.
A negative phase angle means the current leads the voltage — this is characteristic of capacitive loads. A positive angle means current lags voltage — inductive loads. Generators, transformers, and utility meters distinguish leading vs. lagging, but power factor magnitude (|cos θ|) is always positive.
At exactly 90°, the load is purely reactive — an ideal inductor or capacitor. Real power P = 0, and tan θ = Q/P = Q/0 → ∞. In practice, no real load is perfectly reactive, so this is a theoretical limit. The calculator returns Infinity to flag this condition.
To correct power factor from PF₁ to PF₂, the required capacitive reactive power is: Q_cap = P × (tan θ₁ − tan θ₂). Use this calculator to find tan θ₁ (existing load angle) and tan θ₂ (target angle), then multiply the difference by the load's real power in kW to get the required kVAR of capacitor banks.
Yes. The trigonometric relationships are the same for three-phase balanced circuits. The phase angle between line voltage and line current determines power factor in three-phase systems exactly as in single-phase. Common three-phase angles include 30° (phase shift between line and phase voltages in star/delta configurations) and the load power factor angle.
In a balanced three-phase system, the three phasors are separated by 120°. The geometric relationships between line and phase quantities introduce 30° and 60° angles. For example, line voltage in a star (Y) connection leads the phase voltage by 30°. Delta-connected transformer windings introduce 30° phase shifts that must be accounted for in protective relay settings.
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