Geometrie & Funktionen · AP Precalculus Unit 3 (Trigonometric & Polar Functions), A-Level Maths (Trigonometry & Radians), High School Physics (Harmonic Motion).
Einheitskreis & Trigonometrie-Simulator
Trigonometrie am Einheitskreis & Entstehung von Sinus- und Kosinuskurven
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- Echtzeit-Herleitung
- Projektormodus
- A4-Arbeitsblatt
Geometrie-Demofenster● 60 FPS
Experimentelle Steuerungsparameter
°
rad/s
°
an/aus
Echtzeit-Herleitung & Kennzahlen
Sinus sin(θ)
--
Y-coordinate on unit circle
Kosinus cos(θ)
--
X-coordinate on unit circle
Periode der Welle T
--
T = 2π / ω
Wert im Bogenmaß
--
Radians = deg × π / 180
Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Typische Beispielaufgaben & Prüfungsfallen
Wave Transformation: Convert y = sin(x) to y = 2sin(2x + π/3)
Aufgabe: 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.
Analysestrategie: 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!
Schrittweise Herleitung:
- 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.
⚠️ Häufige Fehlvorstellung: Common pitfall: shifting by φ instead of φ/ω when scaling precedes shifting; the horizontal displacement operates strictly on x.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. For angle θ = 150°, what are the values of sin(150°) and cos(150°)?
Answer Key: sin(150°) = 0.5, cos(150°) = -√3/2 ≈ -0.866
Q2. Find the amplitude, period, and initial phase for y = 3sin(4x - π/2).
Answer Key: Amplitude A = 3, Period T = π/2, Initial phase φ = -π/2
Häufige Fragen & typische Missverständnisse
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.