Geometria interativa e funções · AP Precalculus Unit 3 (Trigonometric & Polar Functions), A-Level Maths (Trigonometry & Radians), High School Physics (Harmonic Motion).
Visualizador do círculo trigonométrico e ondas senoidais
Visualizador do círculo unitário e funções trigonométricas: seno, cosseno e tangente
- No navegador
- Sem cadastro
- Dedução ao vivo
- Modo projetor
- Folha de exercícios A4
Janela de demonstração geométrica● 60 FPS
Variáveis de controle do experimento
°
rad/s
°
sim/não
Deduções e métricas em tempo real
Seno sin(θ)
--
Y-coordinate on unit circle
Cosseno cos(θ)
--
X-coordinate on unit circle
Período da onda T
--
T = 2π / ω
Valor em radianos
--
Radians = deg × π / 180
Dedução de fórmulas em tempo real
Fórmula fundamental:
--Substituição de valores:
--Conclusão: --
Exemplos clássicos e armadilhas de exame
Wave Transformation: Convert y = sin(x) to y = 2sin(2x + π/3)
Problema: 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.
Estratégia analítica: 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!
Derivação passo a passo:
- 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.
⚠️ Erro conceitual comum: Common pitfall: shifting by φ instead of φ/ω when scaling precedes shifting; the horizontal displacement operates strictly on x.
Folha de investigação em aula e exercícios
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
Perguntas frequentes e erros comuns
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.