Amplitude and Period Calculator

Calculate amplitude, period, phase shift, and vertical shift of trigonometric functions. Essential trigonometry tool for students and educators.

Settings
Angle Unit:
Radians
Degrees

General Sinusoidal Function: y = A·sin(B·x + C) + D or y = A·cos(B·x + C) + D

Where: A = amplitude, Period = 2π/|B|, Phase shift = -C/B, Vertical shift = D

Select the type of trigonometric function to analyze
Number of periods to display on the graph
Vertical stretch factor
Affects period: Period = 2π/|B|
Horizontal shift inside function
Moves graph up or down
Advanced Options
Display frequency in radians or cycles per unit
Number of decimal places for results
Sum of multiple trigonometric functions (coming soon)
Calculating...

Understanding Amplitude and Period

In trigonometry, sinusoidal functions (sine and cosine) are characterized by several key parameters that determine their shape and position on a graph.

General Sinusoidal Function Form:

y = A·sin(B·x + C) + D

or

y = A·cos(B·x + C) + D

where each parameter affects the graph in a specific way.

Parameter Definitions

Parameter Symbol Effect on Graph Formula
Amplitude A Vertical stretch/compression |A| = (Max - Min)/2
Period P Horizontal length of one cycle P = 2π/|B| (sin/cos)
P = π/|B| (tan)
Frequency B Number of cycles in 2π units f = |B|/(2π)
Phase Shift -C/B Horizontal translation Horizontal shift = -C/B
Vertical Shift D Vertical translation Midline y = D

Angle Units: Radians vs Degrees

1

Radians: The natural unit for trigonometric functions in calculus. One full circle = 2π radians. Most mathematical formulas use radians.

2

Degrees: More intuitive for everyday use. One full circle = 360°. Common in geometry and practical applications.

3

Conversion: π radians = 180°, so to convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.

Applications of Sinusoidal Functions

  • Physics: Modeling simple harmonic motion (pendulums, springs)
  • Engineering: Analyzing alternating current (AC) circuits
  • Audio Engineering: Representing sound waves and frequencies
  • Signal Processing: Fourier analysis and signal decomposition
  • Astronomy: Modeling planetary motion and tidal patterns
  • Economics: Seasonal business cycles and trends

Enhanced Calculator Features:

  • Supports radians and degrees for angle input
  • Enhanced function parser with better support for complex expressions
  • Tangent function support with proper period calculation (π/|B|)
  • Export results as text or image
  • Show/hide key points on the graph
  • Animated wave visualization
  • Improved phase shift explanations

Frequently Asked Questions

Period is the time (or horizontal distance) it takes for one complete cycle of the wave. Frequency is the number of cycles that occur in a unit of time (or distance). They are reciprocals of each other: frequency = 1/period. For sinusoidal functions, the period is 2π/|B| and the angular frequency is |B|.

To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. For example: 90° = 90 × π/180 = π/2 radians. The calculator can handle both units automatically when you select the appropriate setting.

For tangent functions of the form y = A·tan(B·x + C) + D, the period is π/|B|, not 2π/|B|. This is because the tangent function repeats every π radians, unlike sine and cosine which repeat every 2π radians. The calculator automatically adjusts the period calculation based on the function type.

A negative B value causes the function to be reflected horizontally. The period calculation uses |B| (absolute value), so the period remains positive. The phase shift calculation -C/B may produce a non-intuitive result, but the calculator provides an explanation of this. Mathematically, sin(-x) = -sin(x) and cos(-x) = cos(x).

Currently, the calculator focuses on single trigonometric functions. Composite functions (sums of multiple trig functions) don't have a single amplitude or period in the traditional sense. However, we're working on adding support for Fourier analysis which can handle such functions. For now, you can analyze each component separately.