📊 Lesson 3: Trig Functions & Graphs

Amplitude, period, phase shift — the language of waves and oscillations

🎯 Learning Objectives

• Graph y = A sin(Bx + C) + D and y = A cos(Bx + C) + D
• Identify amplitude, period, phase shift, vertical shift
• Write equations from graphs and verbal descriptions
• Connect to physics: wave equations, SHM position/velocity/acceleration

🔑 Key Concepts

Standard Form: y = A sin(ωx + φ) + D

Parameter	Name	Physics Meaning	Effect on Graph
A	Amplitude	Max displacement	Vertical stretch by |A|
ω	Angular frequency	Rad/s (oscillations)	Period = 2π/ω
φ	Phase shift	Initial phase	Horizontal shift by -φ/ω
D	Vertical shift	Equilibrium position	Shift up/down by D

Key Relationships

• Period T = 2π/ω = 1/f
• Frequency f = ω/2π
• Wave speed v = fλ = ωλ/2π = ω/k (where k = 2π/λ is wave number)

Sine vs Cosine

cos θ = sin(θ + π/2) — cosine is sine shifted left by 90° (π/2).

In physics, choose based on initial conditions:

• x = A cos(ωt) — starts at maximum (released from rest at amplitude)
• x = A sin(ωt) — starts at equilibrium moving positive

📐 Worked Examples

Example 1: Analyzing a Wave

y(x,t) = 0.02 sin(4πx - 20πt) (SI units)

Standard form: y = A sin(kx - ωt)
A = 0.02 m = 2 cm (amplitude)
k = 4π rad/m  →  λ = 2π/k = 2π/(4π) = 0.5 m
ω = 20π rad/s  →  f = ω/2π = 10 Hz, T = 0.1 s
v = fλ = 10 × 0.5 = 5 m/s
            

Example 2: SHM Position, Velocity, Acceleration

Mass on spring: x = 0.1 cos(5t) (meters, seconds)

x = 0.1 cos(5t)
v = dx/dt = -0.1(5) sin(5t) = -0.5 sin(5t)
a = dv/dt = -0.5(5) cos(5t) = -2.5 cos(5t) = -ω²x
A = 0.1 m, ω = 5 rad/s
v_max = Aω = 0.5 m/s
a_max = Aω² = 2.5 m/s²
T = 2π/5 ≈ 1.26 s
            

Example 3: Phase Shift from Initial Conditions

At t=0, x=0.05m, v=0.3m/s positive. ω=4 rad/s. Find A, φ for x = A sin(ωt + φ).

x(0) = A sin φ = 0.05
v(0) = Aω cos φ = 0.3  →  A(4) cos φ = 0.3  →  A cos φ = 0.075

Divide: tan φ = 0.05/0.075 = 2/3  →  φ = tan⁻¹(2/3) ≈ 0.588 rad
A = 0.05/sin φ ≈ 0.090 m
x = 0.090 sin(4t + 0.588)
            

🏋️ Practice Problems

1. y = 3 sin(2x). Find A, T, f, ω.
2. y = 2 cos(πx - π/3). Find A, λ, phase shift.
3. Wave: λ=0.4m, f=5Hz, A=0.01m. Write equation y(x,0).
4. SHM: x = 0.2 sin(10t + π/4). Find x_max, v_max, a_max, T.
5. At t=0, x=0, v>0. Which form: A sin(ωt) or A cos(ωt)?
6. Two waves: y₁ = sin(x), y₂ = sin(x + π/2). What is y₁ + y₂? (amplitude, phase)

✅ Answers

Click to reveal answers

1) A=3, ω=2, T=π, f=1/π 2) A=2, λ=2, shift=1/3 right 3) y=0.01 sin(5πx) or cos

4) x_max=0.2, v_max=2, a_max=20, T=0.2π 5) sin (starts at 0, going +) 6) A=√2, phase=π/4

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