Spatial Geometry & Functions · AP Precalculus Unit 3 (Trigonometric & Polar Functions), A-Level Maths (Trigonometry & Radians), High School Physics (Harmonic Motion).

Unit Circle & Trigonometric Wave Visualizer

Where do sine waves come from? They are literally the unwound vertical projections of a point travelling around the unit circle. Watch the height vector sweep across and paint the classic wave in real time.

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angle_deg

amp_A

omega_w

rad/s

phi_deg

show_trig_lines

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Live Derivations & Metrics

val_sin

Y-coordinate on unit circle

val_cos

X-coordinate on unit circle

period_T

T = 2π / ω

angle_rad

Radians = deg × π / 180

Real-Time Formula Derivation

Governing Formula: --

Substitution: --

Regime: --

Worked Examples & Exam Question Deep Dive

Wave Transformation: Convert y = sin(x) to y = 2sin(2x + π/3)

Problem: Detail how to transform y = sin(x) into y = 2sin(2x + π/3) via translation and scaling, clarifying differences between shifting first vs scaling first.

Analytical Strategy: Method 1 (Shift then scale): Shift left by π/3 => sin(x + π/3); scale x by 1/2 => sin(2x + π/3); scale y by 2 => 2sin(2x + π/3). Method 2 (Scale then shift): Scale x by 1/2 => sin(2x); rewrite as 2(x + π/6), requiring shift left by only π/6!

Step-by-Step Derivation:

Step 1: Rewrite target in factored form: y = 2sin[2(x + π/6)].
Step 2: Amplitude scaling: stretch vertically by factor of 2 (A=2).
Step 3: Frequency scaling: compress horizontally by factor of 1/2 (period becomes π).
Step 4: Phase shift caveat: if scaling was applied first, the horizontal shift is φ/ω = (π/3)/2 = π/6 to the left.

⚠️ Common Misconception Pitfall: Common pitfall: shifting by φ instead of φ/ω when scaling precedes shifting; the horizontal displacement operates strictly on x.

Student Inquiry Worksheet & Practice Problems

Q1. For angle θ = 150°, what are the values of sin(150°) and cos(150°)?

Q2. Find the amplitude, period, and initial phase for y = 3sin(4x - π/2).

Frequently Asked Questions & Misconceptions

What is the tangent line on the unit circle?

Draw the tangent line x=1 at (1,0). The terminal ray intersects this line at (1, tan θ). As θ approaches 90°, the intersection recedes to infinity, illustrating why tan(90°) is undefined.