Enter values to verify or solve common trigonometric identities.
sin(A + B)
sin(A – B)
cos(A + B)
cos(A – B)
tan(A + B)
tan(A – B)
sin(2A)
cos(2A)
tan(2A)
Enter values and select an identity.
Understanding and Solving Trigonometric Identities
Trigonometric identities are fundamental equations in trigonometry that hold true for all valid inputs. They are essential tools for simplifying expressions, solving trigonometric equations, and proving other mathematical relationships. This calculator is designed to help you explore and verify some of the most common sum, difference, and double-angle identities.
Key Trigonometric Identities:
Sum and Difference Identities: These identities allow us to find the trigonometric values of the sum or difference of two angles without directly calculating the trigonometric values of the individual angles.
1. Input Angles: Enter the values for Angle A and Angle B in degrees.
2. Select Identity: Choose the trigonometric identity you wish to calculate from the dropdown menu.
3. Calculate: Click the "Calculate" button.
The calculator will output the evaluated result of the chosen trigonometric function for the given angles, based on the selected identity. This can be useful for verifying your manual calculations or quickly obtaining values.
Use Cases:
Educational Tool: Helps students practice and visualize trigonometric identities.
Problem Solving: Aids in simplifying complex trigonometric expressions encountered in calculus, physics, and engineering.
Verification: Allows for quick checking of manually derived results.
Remember that trigonometric functions are periodic and have specific domains and ranges. Ensure your input values are within the expected range for the functions you are working with.
function degreesToRadians(degrees) {
return degrees * Math.PI / 180;
}
function calculateTrigValue(angleInDegrees, func) {
var angleInRadians = degreesToRadians(angleInDegrees);
switch (func) {
case 'sin': return Math.sin(angleInRadians);
case 'cos': return Math.cos(angleInRadians);
case 'tan': return Math.tan(angleInRadians);
default: return NaN;
}
}
function formatResult(value) {
if (isNaN(value)) {
return "Invalid input or undefined result";
}
// Handle potential floating point inaccuracies for common values
if (Math.abs(value) < 1e-10) return "0";
if (Math.abs(value – 1) < 1e-10) return "1";
if (Math.abs(value + 1) < 1e-10) return "-1";
if (Math.abs(value – 0.5) < 1e-10) return "0.5";
if (Math.abs(value + 0.5) < 1e-10) return "-0.5";
if (Math.abs(value – Math.sqrt(2)/2) < 1e-10) return "√2/2";
if (Math.abs(value + Math.sqrt(2)/2) < 1e-10) return "-√2/2";
if (Math.abs(value – Math.sqrt(3)/2) < 1e-10) return "√3/2";
if (Math.abs(value + Math.sqrt(3)/2) < 1e-10) return "-√3/2";
if (Math.abs(value – 1/Math.sqrt(3)) < 1e-10) return "1/√3";
if (Math.abs(value + 1/Math.sqrt(3)) < 1e-10) return "-1/√3";
if (Math.abs(value – Math.sqrt(3)) < 1e-10) return "√3";
if (Math.abs(value + Math.sqrt(3)) < 1e-10) return "-√3";
return parseFloat(value.toFixed(6)); // Limit to 6 decimal places for general numbers
}
function solveTrigIdentity() {
var angleA = parseFloat(document.getElementById("angleA").value);
var angleB = parseFloat(document.getElementById("angleB").value);
var identityType = document.getElementById("identityType").value;
var resultDiv = document.getElementById("result");
if (isNaN(angleA) || isNaN(angleB)) {
resultDiv.textContent = "Please enter valid numbers for angles.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
var sinA = calculateTrigValue(angleA, 'sin');
var cosA = calculateTrigValue(angleA, 'cos');
var tanA = calculateTrigValue(angleA, 'tan');
var sinB = calculateTrigValue(angleB, 'sin');
var cosB = calculateTrigValue(angleB, 'cos');
var tanB = calculateTrigValue(angleB, 'tan');
var calculatedValue;
var identityDescription = "";
switch (identityType) {
case "sinSum":
identityDescription = "sin(A + B)";
calculatedValue = sinA * cosB + cosA * sinB;
break;
case "sinDiff":
identityDescription = "sin(A – B)";
calculatedValue = sinA * cosB – cosA * sinB;
break;
case "cosSum":
identityDescription = "cos(A + B)";
calculatedValue = cosA * cosB – sinA * sinB;
break;
case "cosDiff":
identityDescription = "cos(A – B)";
calculatedValue = cosA * cosB + sinA * sinB;
break;
case "tanSum":
identityDescription = "tan(A + B)";
if (1 – tanA * tanB === 0) { // Avoid division by zero
calculatedValue = Infinity; // Or handle as undefined
} else {
calculatedValue = (tanA + tanB) / (1 – tanA * tanB);
}
break;
case "tanDiff":
identityDescription = "tan(A – B)";
if (1 + tanA * tanB === 0) { // Avoid division by zero
calculatedValue = Infinity; // Or handle as undefined
} else {
calculatedValue = (tanA – tanB) / (1 + tanA * tanB);
}
break;
case "sinDouble":
identityDescription = "sin(2A)";
calculatedValue = 2 * sinA * cosA;
break;
case "cosDouble":
identityDescription = "cos(2A)";
// Using the first form for calculation, others are equivalent
calculatedValue = cosA * cosA – sinA * sinA;
break;
case "tanDouble":
identityDescription = "tan(2A)";
if (1 – tanA * tanA === 0) { // Avoid division by zero
calculatedValue = Infinity; // Or handle as undefined
} else {
calculatedValue = (2 * tanA) / (1 – tanA * tanA);
}
break;
default:
resultDiv.textContent = "Invalid identity selected.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
var formattedResult = formatResult(calculatedValue);
if (formattedResult.includes("Invalid") || formattedResult.includes("undefined")) {
resultDiv.textContent = `Result for ${identityDescription}: ${formattedResult}`;
resultDiv.style.backgroundColor = "#ffc107"; // Warning yellow
} else {
resultDiv.textContent = `Result for ${identityDescription}: ${formattedResult}`;
resultDiv.style.backgroundColor = "var(–success-green)"; // Success green
}
}