Find The Period And Amplitude Of The Function Calculator

Enter the coefficients 'A' and 'B' from your trigonometric function (e.g., y = A sin(Bx), y = A cos(Bx)) to find its amplitude and period. Our period and amplitude of the function calculator will do the work!

Function Graph

Graph of the function y = A sin(Bx) or y = A cos(Bx) over two periods.

Table of Values

x	y = A sin(Bx) / A cos(Bx)

Table showing x and y values for the function over one period at key intervals.

What is Period and Amplitude of a Function?

The period and amplitude of a function, specifically trigonometric functions like sine (sin) and cosine (cos), describe key characteristics of its graph. These functions model periodic phenomena like waves, oscillations, and cycles.

Amplitude: The amplitude is half the distance between the maximum and minimum values of the function. It represents the "height" of the wave from its central axis. For functions of the form y = A sin(Bx) or y = A cos(Bx), the amplitude is |A|.

Period: The period is the smallest interval of x after which the function's values start to repeat. It's the length of one complete cycle of the wave. For y = A sin(Bx) or y = A cos(Bx), the period is 2π / |B| (or 360° / |B| if working in degrees).

This period and amplitude of the function calculator helps you quickly find these values for functions like `y = A sin(Bx)` or `y = A cos(Bx)`.

Who should use it? Students studying trigonometry, engineers, physicists, and anyone working with wave phenomena or periodic signals will find this calculator useful.

Common Misconceptions:

• The amplitude is the peak value, not peak-to-peak.
• The period is determined by 'B', not 'A'. 'A' only affects the amplitude.
• Phase shift (from a 'C' term like `sin(Bx + C)`) and vertical shift (from a 'D' term like `sin(Bx) + D`) do not change the amplitude or period, though they shift the graph. This calculator focuses on the basic form `A sin(Bx)` and `A cos(Bx)`.

Period and Amplitude of the Function Formula and Mathematical Explanation

For a trigonometric function of the form:

y = A sin(Bx + C) + D or y = A cos(Bx + C) + D

The amplitude is given by the absolute value of A:

Amplitude = |A|

The period is calculated based on the absolute value of B:

Period = 2π / |B| (in radians)

Period = 360° / |B| (in degrees)

This period and amplitude of the function calculator uses the radian formula.

Step-by-step derivation:

1. The basic sine and cosine functions (sin(x), cos(x)) have an amplitude of 1 and a period of 2π.
2. Multiplying by A (as in A sin(x)) scales the function vertically, so the maximum value becomes |A| and the minimum becomes -|A|, making the amplitude |A|.
3. Replacing x with Bx (as in sin(Bx)) horizontally compresses or stretches the graph. For the function to complete one cycle, Bx must go from 0 to 2π. This means x goes from 0 to 2π/B. Thus, the period is 2π/|B|. The absolute value is used because period is always positive.

Variables Table:

Variable	Meaning	Unit	Typical Range
A	Amplitude factor/multiplier	Unitless (or units of y)	Any real number
B	Frequency factor (related to angular frequency)	Radians per unit of x (or unitless if x is angle)	Any non-zero real number
Amplitude	Half the peak-to-peak height	Units of y	Non-negative real number
Period	Length of one cycle along x-axis	Units of x	Positive real number

Variables used in the period and amplitude of the function calculator.

Practical Examples (Real-World Use Cases)

Example 1: Sound Wave

A simple sound wave can be modeled by `y = 0.5 sin(440 * 2π * t)`, where y is the pressure variation and t is time in seconds. Here, A = 0.5 and B = 440 * 2π.

• Amplitude |A| = |0.5| = 0.5 (related to loudness)
• Period 2π / |B| = 2π / (440 * 2π) = 1/440 seconds (the reciprocal is the frequency, 440 Hz, which is the A note above middle C)

Using our period and amplitude of the function calculator with A=0.5 and B=880π (approx 2764.6), you'd get these results.

Example 2: Alternating Current (AC)

The voltage in an AC circuit might be given by `V(t) = 170 sin(120πt)`, where V is voltage and t is time in seconds. Here, A = 170 and B = 120π.

• Amplitude |A| = |170| = 170 volts (peak voltage)
• Period 2π / |B| = 2π / (120π) = 1/60 seconds (corresponding to a frequency of 60 Hz)

The period and amplitude of the function calculator can quickly confirm these for A=170 and B=120π (approx 376.99).

How to Use This Period and Amplitude of the Function Calculator

1. Enter Coefficient A: In the "Coefficient A" field, input the value of 'A' from your function `y = A sin(Bx)` or `y = A cos(Bx)`. This determines the amplitude.
2. Enter Coefficient B: In the "Coefficient B" field, input the value of 'B'. This value must be non-zero as it determines the period.
3. Select Function Type: Choose 'sin' or 'cos' based on your function.
4. Calculate: Click the "Calculate" button (or results update as you type).
5. View Results: The calculator will display:
   • The calculated Amplitude (|A|).
   • The calculated Period (2π / |B|).
   • The Angular Frequency |B|.
   • The standard Frequency |B| / 2π.
6. See the Graph and Table: A graph of the function over two periods and a table of values over one period will be generated.
7. Reset: Click "Reset" to clear inputs and results to default values.
8. Copy Results: Click "Copy Results" to copy the main results and inputs to your clipboard.

This period and amplitude of the function calculator makes finding these properties straightforward.

Key Factors That Affect Period and Amplitude Results

1. Value of A: The absolute value of 'A' directly determines the amplitude. A larger |A| means a taller wave, a smaller |A| means a shorter wave.
2. Sign of A: If A is negative, the function is reflected across the x-axis, but the amplitude (|A|) remains positive.
3. Value of B: The absolute value of 'B' inversely affects the period. A larger |B| means a shorter period (more cycles in a given interval, higher frequency), and a smaller |B| means a longer period (fewer cycles, lower frequency).
4. B being non-zero: 'B' cannot be zero because division by zero is undefined. If B=0, the function becomes constant (y = A sin(C) + D), and the concept of a period as defined here doesn't apply.
5. Units of B and x: If x is in radians, B is in radians per unit of x, and the period is 2π/|B|. If x were in degrees, B would be in degrees per unit of x, and the period would be 360/|B|. This calculator assumes radians.
6. Presence of C and D: While this calculator focuses on `A sin(Bx)` and `A cos(Bx)`, in the general form `A sin(Bx + C) + D`, 'C' causes a phase shift (horizontal shift) and 'D' causes a vertical shift. These do NOT change the amplitude or period calculated by this period and amplitude of the function calculator but do shift the graph.

Frequently Asked Questions (FAQ)

Q1: What if A is negative in the period and amplitude of the function calculator?

A1: If A is negative, the amplitude is still positive (|A|). The negative sign means the function is reflected vertically compared to when A is positive, but the height of the wave remains the same.

Q2: What happens if B is zero?

A2: If B is zero, the function becomes constant (e.g., y = A sin(C) + D), and the period is undefined in the context of `2π/|B|`. Our calculator requires B to be non-zero.

Q3: How do I find the period if the function uses degrees?

A3: If your B value is given with x in degrees (e.g., sin(2x°)), the period is 360°/|B|. This calculator uses radians (2π/|B|).

Q4: Does phase shift (C) or vertical shift (D) affect period or amplitude?

A4: No. For y = A sin(Bx + C) + D, the amplitude is |A| and the period is 2π/|B|, regardless of C and D.

Q5: What is the relationship between period and frequency?

A5: Frequency (f) is the reciprocal of the period (T): f = 1/T. Angular frequency (ω or |B|) is 2πf.

Q6: Can this calculator handle tan(Bx)?

A6: No, this calculator is specifically for sin and cos functions. The tangent function (tan(Bx)) has a period of π/|B| and its amplitude is undefined as it goes to infinity.

Q7: What are the units of amplitude and period?

A7: The amplitude has the same units as the function's output (y). The period has the same units as the input variable (x or t).

Q8: How does the period and amplitude of the function calculator graph the function?

A8: It plots y = A sin(Bx) or y = A cos(Bx) for x values ranging from 0 up to two periods, showing two full cycles of the wave based on the calculated amplitude and period.

Related Tools and Internal Resources

• Frequency Calculator: If you know the period, find the frequency, and vice-versa.
• Wavelength Calculator: For waves traveling at a certain speed, relate wavelength, frequency, and speed.
• Trigonometry Basics: Learn more about sine, cosine, and other trigonometric functions.
• Graphing Calculator: Plot more complex functions and explore their properties.
• Angular Frequency Calculator: Calculate angular frequency from frequency or period.
• Phase Shift Calculator: Calculate the phase shift for functions like `A sin(Bx + C) + D`.