Trigonometric Function Graphing Calculator
Enter the parameters for your trigonometric function (e.g., y = A * sin(B*x + C) + D or y = A * cos(B*x + C) + D).
Sine (sin)
Cosine (cos)
(Period = 2π / |B|)
(Shift left by -C/B)
Calculation Results
(Results will appear here after calculation)
Understanding Trigonometric Functions and Graphing
Trigonometric functions like sine and cosine are fundamental in mathematics, physics, engineering, and many other fields. They describe periodic phenomena, such as waves, oscillations, and circular motion. The general form of a transformed sine or cosine function is often represented as:
y = A * f(B*x + C) + D
Where:
frepresents either the sine (sin) or cosine (cos) function.Ais the Amplitude: This determines the maximum displacement or height of the wave from its midline. A larger|A|means a taller wave.Bis the Period Factor: This affects the horizontal stretch or compression of the graph. The period (the length of one complete cycle) is calculated asPeriod = 2π / |B|. A larger|B|results in a shorter period (more cycles in a given interval).Cis the Phase Shift (or horizontal shift): This shifts the graph horizontally. A positiveCshifts the graph to the left, and a negativeCshifts it to the right. The shift amount is effectively-C/B.Dis the Vertical Shift: This shifts the graph up or down. A positiveDshifts the graph upwards, and a negativeDshifts it downwards. This also represents the midline of the function, which is the liney = D.
Use Cases for Trigonometric Functions:
Trigonometric functions are used to model a vast array of real-world phenomena:
- Physics: Describing simple harmonic motion (e.g., pendulums, springs), wave phenomena (sound waves, light waves, electromagnetic waves), alternating current (AC) circuits.
- Engineering: Signal processing, control systems, structural analysis, robotics.
- Mathematics: Calculus, geometry, complex numbers, Fourier analysis.
- Computer Graphics: Animations, procedural generation of textures, simulations.
- Other Fields: Modeling population dynamics, analyzing astronomical cycles, understanding music theory.
How This Calculator Works:
This calculator allows you to input the parameters (Amplitude A, Period Factor B, Phase Shift C, and Vertical Shift D) for a sine or cosine function. It then calculates key characteristics of the graph, such as the period and the midline, and provides example evaluations of the function at specific points. This helps in understanding how each parameter influences the shape and position of the trigonometric graph.
Calculation Results
'; // Clear previous results // — Input Validation — if (isNaN(amplitude) || isNaN(periodFactor) || isNaN(phaseShift) || isNaN(verticalShift)) { resultDiv.innerHTML += 'Please enter valid numerical values for all parameters.'; return; } // — Calculations — var period = (periodFactor === 0) ? Infinity : (2 * Math.PI) / Math.abs(periodFactor); var midline = verticalShift; var maxAmplitude = amplitude; var minAmplitude = -amplitude; var phaseShiftAmount = (periodFactor === 0) ? Infinity : (-phaseShift / periodFactor); // Example evaluations var x_values = [0, period / 4, period / 2, 3 * period / 4, period]; var evaluations = []; for (var i = 0; i < x_values.length; i++) { var x = x_values[i]; if (x === Infinity) continue; // Skip if period is infinite var argument = periodFactor * x + phaseShift; var y_value; if (functionType === "sin") { y_value = amplitude * Math.sin(argument) + verticalShift; } else { // cos y_value = amplitude * Math.cos(argument) + verticalShift; } evaluations.push({ x: x.toFixed(3), y: y_value.toFixed(3) }); } // — Display Results — resultDiv.innerHTML += 'Function Type: ' + functionType.toUpperCase() + "; resultDiv.innerHTML += 'Amplitude (A): ' + amplitude + ''; resultDiv.innerHTML += 'Period Factor (B): ' + periodFactor + ''; resultDiv.innerHTML += 'Phase Shift (C): ' + phaseShift + ''; resultDiv.innerHTML += 'Vertical Shift (D): ' + verticalShift + ''; resultDiv.innerHTML += 'Calculated Period: ' + (period === Infinity ? "Infinite (B=0)" : period.toFixed(3)) + ''; resultDiv.innerHTML += 'Calculated Midline: y = ' + midline.toFixed(3) + ''; resultDiv.innerHTML += 'Maximum y-value: ' + (midline + maxAmplitude).toFixed(3) + ''; resultDiv.innerHTML += 'Minimum y-value: ' + (midline + minAmplitude).toFixed(3) + ''; resultDiv.innerHTML += 'Horizontal Shift Amount (-C/B): ' + (phaseShiftAmount === Infinity ? "N/A (B=0)" : phaseShiftAmount.toFixed(3)) + ''; if (evaluations.length > 0) { resultDiv.innerHTML += 'Example Evaluations (at key points of the cycle):
- ';
for (var j = 0; j < evaluations.length; j++) {
resultDiv.innerHTML += '
- For x = ' + evaluations[j].x + ', y = ' + evaluations[j].y + ' '; } resultDiv.innerHTML += '