Verify the Identity Calculator

Trigonometric Identity Verifier

This calculator helps you numerically verify common trigonometric identities for a given angle. Select an identity, input an angle value, and choose its unit to see if the Left Hand Side (LHS) and Right Hand Side (RHS) of the identity evaluate to the same value.

— Choose an Identity — Pythagorean Identity: sin²(x) + cos²(x) = 1 Tangent Identity: tan(x) = sin(x) / cos(x) Double Angle Sine: sin(2x) = 2sin(x)cos(x) Double Angle Cosine: cos(2x) = cos²(x) – sin²(x) Secant Identity: sec²(x) = 1 + tan²(x) Cosecant Identity: csc²(x) = 1 + cot²(x) Cotangent Identity: cot(x) = cos(x) / sin(x)
function calculateIdentity() { var identitySelect = document.getElementById("identitySelect"); var selectedIdentity = identitySelect.value; var angleInput = document.getElementById("angleValue"); var angleValue = parseFloat(angleInput.value); var unitDegrees = document.getElementById("unitDegrees").checked; var resultDiv = document.getElementById("identityResult"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(angleValue)) { resultDiv.innerHTML = "Please enter a valid number for the angle."; return; } if (selectedIdentity === "") { resultDiv.innerHTML = "Please select an identity to verify."; return; } var angleRad = angleValue; if (unitDegrees) { angleRad = angleValue * Math.PI / 180; } var lhs, rhs; var status = ""; var tolerance = 1e-9; // For floating point comparison try { switch (selectedIdentity) { case "pythagorean": lhs = Math.pow(Math.sin(angleRad), 2) + Math.pow(Math.cos(angleRad), 2); rhs = 1; break; case "tangent": if (Math.abs(Math.cos(angleRad)) < tolerance) { throw new Error("Tangent is undefined at this angle (cosine is zero)."); } lhs = Math.tan(angleRad); rhs = Math.sin(angleRad) / Math.cos(angleRad); break; case "doubleAngleSine": lhs = Math.sin(2 * angleRad); rhs = 2 * Math.sin(angleRad) * Math.cos(angleRad); break; case "doubleAngleCosine": lhs = Math.cos(2 * angleRad); rhs = Math.pow(Math.cos(angleRad), 2) – Math.pow(Math.sin(angleRad), 2); break; case "secant": if (Math.abs(Math.cos(angleRad)) < tolerance) { throw new Error("Secant or Tangent is undefined at this angle (cosine is zero)."); } lhs = 1 / Math.pow(Math.cos(angleRad), 2); // sec^2(x) rhs = 1 + Math.pow(Math.tan(angleRad), 2); // 1 + tan^2(x) break; case "cosecant": if (Math.abs(Math.sin(angleRad)) < tolerance) { throw new Error("Cosecant or Cotangent is undefined at this angle (sine is zero)."); } // Check for tan(angleRad) being zero for cot(x) = 1/tan(x) if (Math.abs(Math.tan(angleRad)) tolerance) { // tan(x) is 0 when sin(x) is 0, but also when x=0, pi, etc. throw new Error("Cotangent is undefined at this angle (tangent is zero)."); } lhs = 1 / Math.pow(Math.sin(angleRad), 2); // csc^2(x) rhs = 1 + Math.pow(1 / Math.tan(angleRad), 2); // 1 + cot^2(x) break; case "cotangent": if (Math.abs(Math.sin(angleRad)) < tolerance) { throw new Error("Cotangent is undefined at this angle (sine is zero)."); } lhs = 1 / Math.tan(angleRad); rhs = Math.cos(angleRad) / Math.sin(angleRad); break; default: resultDiv.innerHTML = "An unexpected error occurred. Please try again."; return; } var angleUnitDisplay = unitDegrees ? "degrees" : "radians"; var angleValueDisplay = angleValue.toFixed(4); // Format for display var lhsFormatted = lhs.toFixed(8); var rhsFormatted = rhs.toFixed(8); if (Math.abs(lhs – rhs) < tolerance) { status = "Verified (within tolerance)"; } else { status = "Not Verified"; } resultDiv.innerHTML = "Selected Identity: " + identitySelect.options[identitySelect.selectedIndex].text + "" + "Angle: " + angleValueDisplay + " " + angleUnitDisplay + "" + "Left Hand Side (LHS) Result: " + lhsFormatted + "" + "Right Hand Side (RHS) Result: " + rhsFormatted + "" + "Verification Status: " + status + ""; } catch (error) { resultDiv.innerHTML = "Error: " + error.message + ""; } } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-container .form-group { margin-bottom: 15px; } .calculator-container label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-container input[type="number"], .calculator-container select { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 16px; } .calculator-container input[type="radio"] { margin-right: 5px; } .calculator-container button { width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; } .calculator-container button:hover { background-color: #0056b3; } .calculator-container .result-section { margin-top: 20px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #e9ecef; color: #333; } .calculator-container .result-section p { margin: 5px 0; line-height: 1.5; } .calculator-container .result-section strong { color: #333; }

Understanding Trigonometric Identities and Their Verification

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables for which both sides of the equation are defined. They are fundamental tools in mathematics, physics, and engineering, simplifying complex expressions, solving equations, and proving other mathematical statements.

What are Trigonometric Identities?

At their core, trigonometric identities are relationships between sine, cosine, tangent, cotangent, secant, and cosecant functions. These relationships stem from the definitions of these functions in a right-angled triangle or on the unit circle. For example, the Pythagorean identity, sin²(x) + cos²(x) = 1, is a direct consequence of the Pythagorean theorem applied to the unit circle.

Why are Identities Important?

  • Simplification: They allow complex trigonometric expressions to be reduced to simpler forms, making calculations easier.
  • Equation Solving: Identities are crucial for solving trigonometric equations, transforming them into forms that are easier to manipulate.
  • Calculus: In calculus, identities are used to simplify integrands and derivatives, making integration and differentiation more manageable.
  • Physics and Engineering: They are applied in wave mechanics, signal processing, electrical engineering, and many other fields where periodic phenomena are modeled.

How to Verify an Identity

Formally verifying an identity involves algebraic manipulation, starting from one side of the equation and transforming it step-by-step into the other side, using known identities and algebraic rules. This process requires a deep understanding of trigonometric relationships and algebraic techniques.

However, a common way to gain confidence in an identity, or to quickly check if a proposed identity might be false, is through numerical verification. This involves substituting a specific angle value into both sides of the equation and checking if they yield the same numerical result. If they don't match, the identity is definitely false. If they do match, it increases confidence, but it does not constitute a formal proof, as an identity must hold true for all valid angles.

Using the Trigonometric Identity Verifier

Our "Trigonometric Identity Verifier" calculator provides a practical way to perform this numerical check. Here's how to use it:

  1. Select Identity: Choose one of the common trigonometric identities from the dropdown menu. Each option clearly states the identity it represents.
  2. Enter Angle Value: Input the angle you wish to test. This can be any real number.
  3. Choose Angle Unit: Specify whether your angle is in degrees or radians. The calculator will convert degrees to radians internally for trigonometric function calculations.
  4. Click "Verify Identity": The calculator will then compute the value of the Left Hand Side (LHS) and the Right Hand Side (RHS) of the selected identity using your provided angle.

Interpreting the Results

The calculator will display the calculated LHS and RHS values, along with a "Verification Status":

  • "Verified (within tolerance)": This means that for the given angle, the numerical values of the LHS and RHS are extremely close (within a small floating-point tolerance). This suggests the identity holds true for this specific angle.
  • "Not Verified": If the LHS and RHS values differ significantly, the identity is not verified for that angle. This could indicate that the identity itself is false, or that there's a specific condition (like division by zero) where the identity is undefined for that angle.
  • "Error: …": The calculator will also catch cases where trigonometric functions are undefined (e.g., tan(90°) or sec(90°)) and provide an appropriate error message.

Examples of Verification:

Let's use the calculator with some realistic numbers:

  1. Pythagorean Identity: sin²(x) + cos²(x) = 1
    • Input: Angle Value = 30, Unit = Degrees
    • Calculation:
      • LHS: sin²(30°) + cos²(30°) = (0.5)² + (0.866025)² = 0.25 + 0.75 = 1
      • RHS: 1
    • Result: LHS (1.00000000) = RHS (1.00000000). Status: Verified.
  2. Tangent Identity: tan(x) = sin(x) / cos(x)
    • Input: Angle Value = 45, Unit = Degrees
    • Calculation:
      • LHS: tan(45°) = 1
      • RHS: sin(45°) / cos(45°) = 0.70710678 / 0.70710678 = 1
    • Result: LHS (1.00000000) = RHS (1.00000000). Status: Verified.
  3. Double Angle Sine: sin(2x) = 2sin(x)cos(x)
    • Input: Angle Value = 60, Unit = Degrees
    • Calculation:
      • LHS: sin(2 * 60°) = sin(120°) = 0.86602540
      • RHS: 2 * sin(60°) * cos(60°) = 2 * 0.86602540 * 0.5 = 0.86602540
    • Result: LHS (0.86602540) = RHS (0.86602540). Status: Verified.

This calculator is a helpful tool for students and professionals alike to quickly check the validity of trigonometric identities for specific numerical inputs, aiding in learning and problem-solving.

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