Calculate properties of composite trigonometric functions y = A sin(ωx + φ) and y = A cos(ωx + φ) including period, range, extrema, symmetry, and monotonic intervals.
Function Parameters
Configure your composite trigonometric function (A > 0, ω > 0)
A > 0
ω > 0
Any real
Current Function:
y = 1 sin(1x + 0)
Parameter Effects
Amplitude (A): Controls the vertical stretch. The function's range becomes [-A, A]. Larger A means taller waves.
Frequency (ω): Controls the horizontal compression. Period = 2π/ω (or 360°/ω). Larger ω means faster oscillation.
Phase (φ): Controls horizontal shift. Positive φ typically shifts left, negative φ shifts right (depending on the form).
函数图像
动态显示 y = 1 sin(1x + 0) 的图像，包括关键点标记
正弦函数
• 圆点表示关键点（极值和零点）
图像显示范围：0 到 4π，参数变化时图像会实时更新
Calculation Methods
Period: For y = A sine(ωx + φ), T = 2π/ω (radians) or 360°/ω (degrees)
Extrema: Solve ωx + φ = key angles where sine reaches ±1
Symmetry: Use standard sine symmetry patterns, adjusted by transformations
Monotonicity: Analyze where derivative is positive/negative using chain rule
Note: All formulas use general form with integer k. The phase shift depends on how you interpret the function form. Results show standard mathematical conventions.