〰️ Sinusoidal Waves & Identities
Every AC voltage, vibration and oscillation in engineering is described by a sinusoidal waveform — amplitude, frequency and phase are the three numbers that define it completely.
2Topics
8Worked Examples
4Eng. Cases
12Practice Qs
〰️Waveform ParametersAmplitude, frequency, phase
➕Combining WavesCompound angle identities
⚙️ApplicationsReal engineering
🎮Quick-Fire GameTest your speed
📝PracticeGraded questions
🎯What you will learn
Identify amplitude, angular frequency, period and phase angle from a sinusoidal equation; sketch and interpret phase-shifted waveforms; and combine two sinusoidal signals into a single equivalent wave using compound angle identities.
💡 v = V_m sin(ωt + φ) packs three physically meaningful numbers into one equation: V_m (how big), ω (how fast), and φ (how shifted in time relative to a reference).
🧭How to use this resource
1. Read the rules banner at the top of each tab
2. Experiment with the interactive tool
3. Check the key vocabulary panel
4. Study the visual diagram, then the worked examples
〰️
Waveform Parameters & Phasors
Three numbers completely define any sinusoidal waveform — amplitude, angular frequency and phase.
🔧Waveform Parameter ExtractorEnter V_m, f, φ and t to evaluate instantaneous value
Enter V_m, f, φ and t, then press Calculate to find ω, the period, and the instantaneous voltage…
📖 Key Vocabulary
Amplitude (V_m) — the maximum value the waveform reaches.
Angular frequency (ω) — the rate of change of phase, in radians per second.
Period (T) — the time for one complete cycle, T=1/f.
Phase angle (φ) — the horizontal shift of the wave relative to a reference sin(ωt) curve.
👁️ Visual aid: the dashed red wave has the same shape as the blue reference wave but starts earlier — that's exactly what a positive phase angle does.
✏️Worked Example 1 — Reading off wave parametersType: Parameter identification
Question For v = 325sin(314t + 0.5), state the amplitude, angular frequency, frequency and phase.
Read directly V_m=325V, ω=314 rad/s, φ=0.5 rad
Find f
f = ω/2π = 314/6.283 ≈ 50 HzAnswer V_m=325V, f=50Hz, φ=0.5 rad (≈28.6°)
⏱️Worked Example 2 — Finding the periodType: Period calculation
Question A waveform has frequency 60Hz. Find its period.
Apply formula
T = 1/f = 1/60Answer T ≈ 0.01667 s = 16.67 ms
📈Worked Example 3 — Instantaneous value at a given timeType: Evaluation
Question For v = 100sin(ωt) with f=50Hz, find v at t=2ms.
Find ω
ω = 2π(50) = 314.16 rad/sSubstitute
v = 100sin(314.16 × 0.002) = 100sin(0.6283 rad)Answer v = 100 × 0.5878 ≈ 58.78 V
⚙️Worked Example 4 — Engineering context (phase difference between signals)Type: Engineering
Question Voltage v=240sin(ωt) and current i=15sin(ωt−0.4) are measured on a circuit. Describe the phase relationship and state whether current leads or lags voltage.
Compare phase angles Voltage has φ=0, current has φ=−0.4 rad
Answer Current lags voltage by 0.4 rad (≈22.9°) — typical of an inductive circuit, where the current's peak occurs after the voltage's peak.
🧠 Common slip: mixing degrees and radians inside the same sin() calculation — always confirm the phase angle's units before adding it to ωt (which is naturally in radians).
➕
Combining Waves & Identities
Two sinusoidal signals of the same frequency always combine into a single equivalent sinusoid.
🔧Wave CombinerCombine a·sinθ + b·cosθ into R·sin(θ+α)
Enter a and b, then press Combine to find R and α for a·sinθ + b·cosθ = R·sin(θ+α)…
📖 Key Vocabulary
Resultant wave — the single sinusoid equivalent to the sum of two or more sinusoids of the same frequency.
R — the resultant amplitude, always found using R=√(a²+b²).
α — the resultant phase angle, found from tanα=b/a with quadrant checked against the signs of a and b.
Compound angle formula — sin(A+B)=sinAcosB+cosAsinB, the identity underlying wave combination.
👁️ Visual aid: the sin and cos components act like perpendicular vectors — their vector sum gives the resultant amplitude R at angle α.
✏️Worked Example 1 — Combining sin and cos termsType: Full combination
Question Express 8sinθ + 6cosθ in the form R sin(θ+α).
Find R
R = √(8²+6²) = √100 = 10Find α
tanα = 6/8 = 0.75 → α = 36.87°Answer 8sinθ + 6cosθ = 10sin(θ + 36.87°)
🔀Worked Example 2 — A negative cos coefficientType: Quadrant care
Question Express 5sinθ − 12cosθ in the form R sin(θ+α).
Find R
R = √(5²+12²) = √169 = 13Find α
tanα = −12/5 = −2.4 → α = −67.38° (a is positive, b is negative → α is negative, in the correct quadrant here)Answer 5sinθ − 12cosθ = 13sin(θ − 67.38°)
🔁Worked Example 3 — Combining two full sinusoidsType: Two sinusoids
Question Combine v₁=50sin(ωt) and v₂=50sin(ωt+90°) into a single sinusoid.
Expand v₂
50sin(ωt+90°) = 50cos(ωt)Total
50sin(ωt) + 50cos(ωt), so a=50, b=50Combine
R=√(50²+50²)=70.71, α=tan⁻¹(50/50)=45°Answer v_total = 70.71sin(ωt + 45°)
⚙️Worked Example 4 — Engineering context (three-phase supply component)Type: Engineering
Question Two vibration components acting on a machine mount are x₁=3sin(ωt) mm and x₂=4cos(ωt) mm. Find the resultant vibration amplitude and phase.
Set up
a=3, b=4Find R
R=√(3²+4²)=5Find α
tanα=4/3=1.333 → α=53.13°Answer Resultant vibration = 5sin(ωt+53.13°) mm — the classic 3-4-5 triangle appears directly in the vibration analysis.
🧠 Common slip: forgetting to check the quadrant of α using the signs of a and b separately — a calculator's tan⁻¹ alone can silently put α in the wrong quadrant.
⚙️
Engineering Applications
Four live scenarios — each with its own calculator. Enter your own numbers.
🔌 AC Voltage Instantaneous Value
Finding the exact voltage at any moment in an AC cycle, used constantly in circuit analysis.
v = V_m sin(2πft + φ)
📟 Try it — Instantaneous Voltage Calculator
Enter V_m, f, φ and t, then press Calculate…
📳 Combining Vibration Sources
Two vibration sources of the same frequency acting on a structure combine into one resultant sinusoid.
R = √(a²+b²), α = tan⁻¹(b/a)
📟 Try it — Vibration Combiner
Enter a and b, then press Calculate…
🔧 Phase Difference Between Signals
Comparing the phase angles of two signals (e.g. voltage and current) to determine lead/lag relationships.
Phase difference = φ₁ − φ₂
📟 Try it — Phase Difference Calculator
Enter both phase angles, then press Calculate…
⏱️ Frequency & Period Conversion
Converting between frequency and period is a routine step in specifying oscillators, sensors and control loops.
T = 1/f, ω = 2πf
📟 Try it — Frequency/Period Converter
Enter the frequency, then press Calculate for T and ω…
🎮
Quick-Fire Sinusoidal Waves Game
12 questions on waveform parameters and combining waves. How many can you get?
⚡ Sinusoidal Waves Challenge
Score: 0Question: 1/12Streak: 0🔥
Press Start to begin!
📝
Practice Questions
Graded from warm-up to challenge. Attempt fully before revealing the working.
Question 1 🥉2 marks
For v=100sin(200t), state the amplitude and angular frequency.
V_m=100, ω=200 rad/s
Question 2 🥉2 marks
Find the frequency for ω=100π rad/s.
f=ω/2π=100π/2π=50 Hz
Question 3 🥉2 marks
Find the period for f=25Hz.
T=1/25=0.04 s
Question 4 🥉2 marks
Find R for a=3sinθ+4cosθ combined into Rsin(θ+α).
R=√(9+16)=√25=5
Question 5 🥈3 marks
Find α for 3sinθ+4cosθ = Rsin(θ+α).
tanα=4/3=1.333
α=53.13°
α=53.13°
Question 6 🥈4 marks
For v=200sin(157t), find v at t=0.01s.
v=200sin(1.57 rad)=200×0.99999
≈200.0 V
≈200.0 V
Question 7 🥈3 marks
Express 5sinθ−5cosθ in the form Rsin(θ+α).
R=√(25+25)=√50=7.071
tanα=−5/5=−1, α=−45°
7.071sin(θ−45°)
tanα=−5/5=−1, α=−45°
7.071sin(θ−45°)
Question 8 🥈4 marks
Two signals have phase angles φ₁=30° and φ₂=−15°. Find the phase difference and state which leads.
Difference = 30−(−15) = 45° — Signal 1 leads Signal 2 by 45°
Question 9 🥇4 marks
A supply voltage v=340sin(314t+0.2) is measured. Find V_m, f and φ (in degrees).
V_m=340V
f=314/(2π)=49.98≈50Hz
φ=0.2 rad × (180/π) = ≈11.46°
f=314/(2π)=49.98≈50Hz
φ=0.2 rad × (180/π) = ≈11.46°
Question 10 🥇5 marks
Combine i₁=10sin(ωt) and i₂=10sin(ωt+120°) into a single sinusoid (expand i₂ first).
i₂=10[sin(ωt)cos120°+cos(ωt)sin120°]=10[sin(ωt)(−0.5)+cos(ωt)(0.866)]
=−5sin(ωt)+8.66cos(ωt)
Total: (10−5)sin(ωt)+8.66cos(ωt) = 5sin(ωt)+8.66cos(ωt)
R=√(25+75)=√100=10
α=tan⁻¹(8.66/5)=60°; i_total=10sin(ωt+60°)
=−5sin(ωt)+8.66cos(ωt)
Total: (10−5)sin(ωt)+8.66cos(ωt) = 5sin(ωt)+8.66cos(ωt)
R=√(25+75)=√100=10
α=tan⁻¹(8.66/5)=60°; i_total=10sin(ωt+60°)
Question 11 🥇5 marks
Two mounting vibrations act as x₁=7sin(ωt)mm and x₂=−24cos(ωt)mm. Find the resultant amplitude and phase.
a=7, b=−24
R=√(49+576)=√625=25
tanα=−24/7=−3.4286, α=−73.74°
Resultant = 25sin(ωt−73.74°) mm
R=√(49+576)=√625=25
tanα=−24/7=−3.4286, α=−73.74°
Resultant = 25sin(ωt−73.74°) mm
Question 12 🥇5 marks
A machine vibrates with x=12sin(ωt)+5cos(ωt) mm. Find the maximum displacement and the time (as a phase angle) at which it first occurs after t=0.
R=√(144+25)=√169=13mm (maximum displacement)
α=tan⁻¹(5/12)=22.62°
Peak occurs when (ωt+22.62°)=90°, i.e. ωt=67.38°
Max = 13mm, first occurring at phase ωt=67.38°
α=tan⁻¹(5/12)=22.62°
Peak occurs when (ωt+22.62°)=90°, i.e. ωt=67.38°
Max = 13mm, first occurring at phase ωt=67.38°