Trigonometric Identity Verifier
This calculator helps you verify trigonometric identities by evaluating both sides of an equation for a given angle. While numerical verification for a single angle does not constitute a formal proof, it can help you check your work, identify potential errors, or disprove false identities.
e.g., Math.sin(x), 1/Math.cos(x), Math.pow(Math.tan(x), 2)
e.g., Math.tan(x), Math.sin(2*x)
How to Use This Calculator:
To use the Trigonometric Identity Verifier, follow these guidelines for entering expressions:
- Angle Variable: Always use
xas the variable for the angle. - Trigonometric Functions:
- Sine:
Math.sin(x) - Cosine:
Math.cos(x) - Tangent:
Math.tan(x)
- Sine:
- Reciprocal Functions:
- Secant (sec(x)):
1/Math.cos(x) - Cosecant (csc(x)):
1/Math.sin(x) - Cotangent (cot(x)):
1/Math.tan(x)
- Secant (sec(x)):
- Powers: Use
Math.pow(base, exponent). For example,sin^2(x)should be entered asMath.pow(Math.sin(x), 2). - Constants: Use
Math.PIfor π (pi). - Operations: Standard arithmetic operators (
+,-,*,/) are supported. Use parentheses()to ensure correct order of operations.
Understanding the Results:
The calculator evaluates both the Left Side Expression and the Right Side Expression for the given angle. It then compares the numerical results:
- If the values are very close (within a small tolerance due to floating-point arithmetic), the calculator will suggest the identity holds true for the given angle.
- If the values are significantly different, the calculator will indicate that the identity does not hold true for the given angle, suggesting the identity is false.
- Remember, verifying an identity for one angle does not prove it for all angles. However, if it fails for even one angle, the identity is definitively false.
Examples:
Example 1: Pythagorean Identity
Identity: sin^2(x) + cos^2(x) = 1
- Angle Value:
30 - Angle Unit:
Degrees - Left Side Expression:
Math.pow(Math.sin(x), 2) + Math.pow(Math.cos(x), 2) - Right Side Expression:
1 - Expected Result: Both sides should evaluate to
1.
Example 2: Double Angle Identity
Identity: sin(2x) = 2sin(x)cos(x)
- Angle Value:
45 - Angle Unit:
Degrees - Left Side Expression:
Math.sin(2*x) - Right Side Expression:
2 * Math.sin(x) * Math.cos(x) - Expected Result: Both sides should evaluate to
1.
Example 3: A False Identity
Identity: sin(x) + cos(x) = 1 (This is generally false)
- Angle Value:
30 - Angle Unit:
Degrees - Left Side Expression:
Math.sin(x) + Math.cos(x) - Right Side Expression:
1 - Expected Result: Left side evaluates to approximately
1.366. Right side is1. They are not equal.
Verification Results:
'; outputHtml += 'Angle (x): ' + angleValue + ' ' + (unitDegrees ? 'Degrees' : 'Radians') + "; outputHtml += 'Left Side Expression:' + expressionLeft + '';
outputHtml += 'Evaluated Left Side: ' + leftResult.toFixed(10) + ";
outputHtml += 'Right Side Expression: ' + expressionRight + '';
outputHtml += 'Evaluated Right Side: ' + rightResult.toFixed(10) + ";
if (Math.abs(leftResult – rightResult) < tolerance) {
outputHtml += 'Conclusion: The identity appears to hold true for this angle (within a small tolerance).';
} else {
outputHtml += 'Conclusion: The identity does NOT hold true for this angle. The difference is ' + Math.abs(leftResult – rightResult).toFixed(10) + '.';
}
resultOutput.innerHTML = outputHtml;
}