Calculate the amplitude and period of a trigonometric function.
Sine (sin)
Cosine (cos)
Tangent (tan)
The 'A' in A*f(B(x-C)) + D.
The 'B' in A*f(B(x-C)) + D. Affects period.
The 'C' in A*f(B(x-C)) + D. Horizontal shift.
The 'D' in A*f(B(x-C)) + D. Vertical shift.
Amplitude: —
Period: —
Understanding Amplitude and Period
In trigonometry and wave phenomena, the amplitude and period are fundamental characteristics that describe the behavior and extent of a periodic function. These concepts are crucial in fields like physics (for waves, oscillations), engineering (signal processing), economics (business cycles), and many other areas where cyclical patterns occur.
Amplitude
The amplitude of a periodic function (like sine or cosine) represents its maximum displacement or the 'height' of the wave from its midline or equilibrium position. It essentially measures the intensity or strength of the wave.
For a function in the standard form:
y = A * f(B(x – C)) + D
where 'f' is a basic trigonometric function (sin, cos, tan),
A is the amplitude. It's the absolute value of the coefficient multiplying the function.
D is the vertical shift, which changes the midline of the function.
The amplitude is calculated as the absolute value of A. If A is positive, the function behaves as expected. If A is negative, the function is reflected across its midline.
For sine and cosine waves, the amplitude is half the difference between the maximum and minimum values of the function:
Amplitude = (Maximum Value – Minimum Value) / 2
For the tangent function, the amplitude is considered infinite or undefined because its range is all real numbers.
Period
The period of a periodic function is the length of one complete cycle or oscillation. It tells us how often a pattern repeats.
In the standard form:
y = A * f(B(x – C)) + D
The period is determined by the coefficient 'B'. The formula for the period depends on the type of trigonometric function:
For Sine (sin) and Cosine (cos) functions: The period is 2π / |B|.
For Tangent (tan) and Cotangent (cot) functions: The period is π / |B|.
The phase shift (C) and vertical shift (D) do not affect the period.
Use Cases
Physics: Describing simple harmonic motion, sound waves, light waves, and electromagnetic radiation. Amplitude relates to loudness/intensity, and period relates to frequency/pitch.
Engineering: Analyzing alternating current (AC) circuits, signal processing, and control systems.
Biology: Modeling population dynamics, heart rhythms, and biological cycles.
Economics: Understanding business cycles and market fluctuations.
This calculator helps you quickly determine these key characteristics for common trigonometric functions, aiding in the analysis of periodic phenomena.
function updateInputLabels() {
var funcType = document.getElementById("functionType").value;
var amplitudeLabel = document.getElementById("amplitudeLabel");
var bValueLabel = document.getElementById("bValueLabel");
if (funcType === "tan") {
amplitudeLabel.innerText = "Amplitude (A):"; // Tangent has undefined amplitude, but we show A for reflection/scaling.
bValueLabel.innerText = "B Value:";
} else {
amplitudeLabel.innerText = "Amplitude (A):";
bValueLabel.innerText = "B Value:";
}
}
function calculateAmplitudePeriod() {
var funcType = document.getElementById("functionType").value;
var amplitudeInput = parseFloat(document.getElementById("amplitudeInput").value);
var bValueInput = parseFloat(document.getElementById("bValueInput").value);
var phaseShiftInput = parseFloat(document.getElementById("phaseShiftInput").value); // Not used in calculation, but included for completeness of form
var verticalShiftInput = parseFloat(document.getElementById("verticalShiftInput").value); // Not used in calculation
var calculatedAmplitudeDisplay = document.getElementById("calculatedAmplitude");
var calculatedPeriodDisplay = document.getElementById("calculatedPeriod");
// Clear previous results
calculatedAmplitudeDisplay.innerText = "–";
calculatedPeriodDisplay.innerText = "–";
var isValid = true;
var errors = [];
if (isNaN(amplitudeInput) || amplitudeInput === null) {
errors.push("Amplitude (A) is required.");
isValid = false;
}
if (isNaN(bValueInput) || bValueInput === null) {
errors.push("B Value is required.");
isValid = false;
}
if (!isValid) {
alert("Please correct the following errors:\n- " + errors.join("\n- "));
return;
}
var calculatedAmplitude;
var calculatedPeriod;
// Calculate Amplitude
if (funcType === "tan") {
calculatedAmplitude = "Undefined (or use |A| for vertical scaling)";
} else {
calculatedAmplitude = Math.abs(amplitudeInput);
}
// Calculate Period
if (bValueInput === 0) {
errors.push("B value cannot be zero for period calculation.");
isValid = false;
}
if (!isValid) {
alert("Please correct the following errors:\n- " + errors.join("\n- "));
return;
}
if (funcType === "sin" || funcType === "cos") {
calculatedPeriod = (2 * Math.PI) / Math.abs(bValueInput);
calculatedPeriod = calculatedPeriod.toFixed(4); // Format to 4 decimal places
} else if (funcType === "tan") {
calculatedPeriod = Math.PI / Math.abs(bValueInput);
calculatedPeriod = calculatedPeriod.toFixed(4); // Format to 4 decimal places
} else {
calculatedPeriod = "N/A";
}
calculatedAmplitudeDisplay.innerText = calculatedAmplitude;
calculatedPeriodDisplay.innerText = calculatedPeriod + (funcType === "tan" ? "" : " units"); // Add units if not tangent
// Add visual cue for undefined amplitude
if (funcType === "tan") {
calculatedAmplitudeDisplay.style.color = "#ffc107"; // Warning color
} else {
calculatedAmplitudeDisplay.style.color = "#28a745";
}
calculatedPeriodDisplay.style.color = "#28a745"; // Success green for period
}
// Initialize labels on page load
window.onload = updateInputLabels;