The Trigonometry Cos Formula: Definition and Phasor Reality
When bench-testing AC circuits or syncing multiple power sources, scalar arithmetic fails. You cannot simply add 120V and 120V to get 240V if the waveforms are out of phase. This is where the trigonometry cos formula—formally known as the Law of Cosines—becomes the most critical mathematical tool in an electrical engineer's or advanced hobbyist's arsenal. The formula defines the relationship between the magnitudes of two vectors (phasors) and the angle between them to find the resultant vector.
The core formula is expressed as:
c² = a² + b² - 2ab·cos(C)
When it applies: This formula applies anytime you need to find the magnitude of a resultant AC voltage or current formed by two sinusoidal sources operating at the exact same frequency but separated by a known phase angle. It is the backbone of 3-phase line-to-line voltage calculations, split-phase inverter syncing, and fault current vector addition.
Assumptions: The formula assumes steady-state sinusoidal waveforms. It also assumes that both phasors a and b share the identical fundamental frequency (e.g., both exactly 60.00 Hz). If frequencies differ, the phase angle C is constantly rotating, and the trigonometry cos formula only yields an instantaneous snapshot, not a stable RMS magnitude.
Symbol Definition and Rearranged Forms
To use the formula on the bench, you must map the mathematical variables to your electrical parameters. Below is the definitive spec sheet for the variables.
| Symbol | Electrical Equivalent | Standard Unit | Definition |
|---|---|---|---|
| a | Phasor 1 Magnitude (e.g., V_phase1) | Volts (RMS) or Amps (RMS) | The magnitude of the first known AC vector. |
| b | Phasor 2 Magnitude (e.g., V_phase2) | Volts (RMS) or Amps (RMS) | The magnitude of the second known AC vector. |
| C | Phase Shift (θ) | Degrees (°) or Radians (rad) | The electrical angle between phasor a and phasor b. |
| c | Resultant Magnitude (e.g., V_line) | Volts (RMS) or Amps (RMS) | The magnitude of the combined or differential resultant vector. |
Rearranged Forms List
Depending on what your multimeter or oscilloscope can measure, you will need to isolate different variables. Here are the algebraic rearrangements solving for each parameter:
- Solving for Resultant (c): c = √(a² + b² - 2ab·cos(C))
- Solving for Phase Angle (C): C = arccos((a² + b² - c²) / 2ab)
- Solving for Phasor 1 (a): Using the quadratic formula on the original equation yields: a = b·cos(C) ± √(c² - b²·sin²(C)). (Note: In electrical practice, you select the positive root that aligns with your physical circuit topology).
- Solving for Phasor 2 (b): b = a·cos(C) ± √(c² - a²·sin²(C))
Worked Bench Problems: Unit Tracking and Magnitude Checks
Abstract math is useless if you drop your units. Let us walk through two common bench scenarios, tracking the units through every intermediate step to ensure the final magnitude makes physical sense.
Problem 1: 3-Phase Wye Line-to-Line Voltage
Scenario: You are measuring a commercial 3-phase wye panel. Phase A (a) and Phase B (b) both read 120V RMS to neutral. The phase shift (C) between them is exactly 120°. Find the line-to-line voltage (c) across the A and B breaker poles.
- Setup: c² = (120V)² + (120V)² - 2(120V)(120V)·cos(120°)
- Square the magnitudes: c² = 14,400 V² + 14,400 V² - 2(14,400 V²)·cos(120°)
- Evaluate the cosine: cos(120°) = -0.5
- Multiply the cross-term: -2(14,400 V²)·(-0.5) = +14,400 V²
- Sum the terms: c² = 14,400 V² + 14,400 V² + 14,400 V² = 43,200 V²
- Take the square root: c = √(43,200 V²) = 207.84 V
Magnitude Check: This perfectly aligns with the nominal 208V 3-phase standard (120V × √3 = 207.84V).
Problem 2: Split-Phase 240V from Dual Inverters
Scenario: You are stacking two 120V off-grid inverters to create a 240V split-phase supply. Inverter 1 (a) outputs 120V. Inverter 2 (b) outputs 120V. They are configured exactly 180° out of phase (C = 180°) to create the split-phase differential.
- Setup: c² = (120V)² + (120V)² - 2(120V)(120V)·cos(180°)
- Evaluate the cosine: cos(180°) = -1
- Multiply the cross-term: -2(14,400 V²)·(-1) = +28,800 V²
- Sum the terms: c² = 14,400 V² + 14,400 V² + 28,800 V² = 57,600 V²
- Take the square root: c = √(57,600 V²) = 240.0 V
Magnitude Check: Because the vectors are perfectly opposed (180°), their differential magnitude is the simple scalar sum (120 + 120 = 240).
Real-World Scenario Walkthrough: The Out-of-Phase Generator Sync
Math on paper is clean; jobsite reality is messy. Here is a documented failure from a 2025 off-grid cabin build where ignoring the trigonometry cos formula led to equipment damage.
The Numbers:
- c² = (122V)² + (118V)² - 2(122V)(118V)·cos(165°)
- c² = 14,884 V² + 13,924 V² - 2(14,396 V²)·(-0.9659)
- c² = 28,808 V² + 27,810 V² = 56,618 V²
- c = √(56,618 V²) = 237.9 V
The Outcome: The multimeter read 237.9V at the pump terminals, which the DIYer assumed was close enough to 240V. However, the pump motor rapidly overheated and tripped its internal thermal overload within three minutes.
What Went Wrong: The trigonometry cos formula reveals that the voltage magnitude was only slightly low (237.9V vs 240V), but it completely masks the phase asymmetry. Because the generators were 15° out of sync (165° instead of 180°), the neutral return path was carrying a massive circulating vector current rather than zero current. The DIYer had bonded the neutrals at the transfer switch. The 15° phase error forced the neutral bus to carry the vector difference of the two hot legs, overheating the neutral wire and creating a severe phase imbalance inside the single-phase pump motor windings. Lesson: Never parallel non-inverter generators for 240V without a dedicated phase-sync lock or a transfer switch that physically isolates the neutrals.
Unit Mistakes and Magnitude Sanity Checks
When plugging values into the trigonometry cos formula, two specific unit mistakes will instantly brick your calculations and lead to dangerous bench assumptions.
The Degree vs. Radian Trap
Your scientific calculator or Python script must be set to the correct angular mode. In electrical power systems, phase angles are universally specified in degrees (e.g., 120° for 3-phase, 180° for split-phase). If your calculator is in Radian mode, cos(120) evaluates to 0.775 instead of -0.5. This will result in a calculated line-to-line voltage of roughly 105V instead of 208V, leading you to believe the panel is severely browned out when it is actually perfectly healthy.
Mixing Peak and RMS Voltages
The formula is mathematically agnostic, but your hardware is not. You must use either entirely RMS values or entirely Peak values for a, b, and c. Standard multimeters (like the Fluke 87V) read True-RMS. Oscilloscopes read Peak-to-Peak or Amplitude (Peak). If a is 120V RMS and b is 170V Peak (which are the same physical waveform), plugging them directly into the formula will yield a nonsensical resultant. Always convert scope readings to RMS (Peak / √2) before applying the formula.
Sanity Check: The Triangle Inequality
What does a realistic answer magnitude look like? The resultant c must always obey the Triangle Inequality Theorem: |a - b| ≤ c ≤ (a + b). If your math yields a resultant voltage higher than the simple sum of the two sources (e.g., getting 260V from two 120V sources), you have made a sign error in your cosine evaluation. Conversely, if c is smaller than the absolute difference between a and b, your phase angle input is physically impossible for the given magnitudes. Use this boundary check every time you verify a phasor calculation on the bench.
For further reading on AC vector mathematics and phasor diagrams, consult the All About Circuits AC Textbook or the Fluke guide on three-phase power fundamentals.
