Verifying Trig Identity Calculator

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Verifying Trig Identity Calculator

Verify if two trigonometric expressions are mathematically equivalent.

Syntax Guide: Use sin(x), cos(x), tan(x), sec(x), csc(x), cot(x). For powers, use ^ or ** (e.g., sin(x)^2).

Understanding Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that holds true for every value of the occurring variables where both sides of the equation are defined. This calculator uses numerical sampling to test if your provided expressions are identical.

Fundamental Identities

  • Pythagorean: sin²(x) + cos²(x) = 1
  • Reciprocal: sec(x) = 1/cos(x), csc(x) = 1/sin(x), cot(x) = 1/tan(x)
  • Quotient: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)

Manual Verification Steps

  1. Work on only one side of the equation (usually the more complex side).
  2. Rewrite all functions in terms of sine and cosine.
  3. Look for algebraic patterns like factoring, common denominators, or FOIL.
  4. Apply Pythagorean identities to simplify squared terms.

Example to Try:

LHS: tan(x) * cos(x)
RHS: sin(x)

function verifyIdentity() { var lhsRaw = document.getElementById('lhsInput').value.trim(); var rhsRaw = document.getElementById('rhsInput').value.trim(); var resDiv = document.getElementById('resultArea'); var statusMsg = document.getElementById('statusMessage'); var detailMsg = document.getElementById('detailMessage'); if (!lhsRaw || !rhsRaw) { alert('Please enter both expressions.'); return; } resDiv.style.display = 'block'; function formatExpression(expr) { var e = expr.toLowerCase(); // Replace functions with Math equivalents e = e.replace(/sin/g, 'Math.sin'); e = e.replace(/cos/g, 'Math.cos'); e = e.replace(/tan/g, 'Math.tan'); // Handle reciprocals e = e.replace(/sec\(([^)]+)\)/g, '(1/Math.cos($1))'); e = e.replace(/csc\(([^)]+)\)/g, '(1/Math.sin($1))'); e = e.replace(/cot\(([^)]+)\)/g, '(1/Math.tan($1))'); // Handle pi e = e.replace(/pi/g, 'Math.PI'); // Handle powers e = e.replace(/\^/g, '**'); return e; } try { var lhsFormatted = formatExpression(lhsRaw); var rhsFormatted = formatExpression(rhsRaw); // Sample points to check (avoid 0 or multiples of pi/2 where tan/sec/csc might be undefined) var testPoints = [0.245, 0.567, 1.12, 2.34, 4.56, -0.78]; var isIdentity = true; var tolerance = 0.0000001; for (var i = 0; i tolerance) { isIdentity = false; break; } } if (isIdentity) { resDiv.style.backgroundColor = '#d4edda'; resDiv.style.border = '1px solid #c3e6cb'; statusMsg.style.color = '#155724'; statusMsg.innerHTML = "✓ Identity Verified!"; detailMsg.innerHTML = "The expressions are mathematically equivalent for all tested values of x."; } else { resDiv.style.backgroundColor = '#f8d7da'; resDiv.style.border = '1px solid #f5c6cb'; statusMsg.style.color = '#721c24'; statusMsg.innerHTML = "✗ Not an Identity"; detailMsg.innerHTML = "The expressions produced different results. Check for sign errors or missing terms."; } } catch (err) { resDiv.style.backgroundColor = '#fff3cd'; resDiv.style.border = '1px solid #ffeeba'; statusMsg.style.color = '#856404'; statusMsg.innerHTML = "Expression Error"; detailMsg.innerHTML = "Ensure you are using standard notation: sin(x), cos(x), etc. Make sure parentheses are closed."; } }

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