Trigonometric Identity Verifier
This tool helps you numerically verify trigonometric identities by evaluating both sides of a proposed identity at a specific angle. While numerical verification can indicate if an identity *might* be true or if it's definitely false for a given angle, it does not constitute a formal mathematical proof. A true proof requires symbolic manipulation using known identities and algebraic rules.
Degrees Radians
Use 'x' for the angle. Functions: sin(x), cos(x), tan(x), asin(x), acos(x), atan(x). Powers: pow(base, exponent). Square root: sqrt(x). Constants: PI, E.
Use 'x' for the angle. Functions: sin(x), cos(x), tan(x), asin(x), acos(x), atan(x). Powers: pow(base, exponent). Square root: sqrt(x). Constants: PI, E.
Understanding Trigonometric Identities
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 in trigonometry and calculus, used for simplifying expressions, solving equations, and proving other mathematical statements.
Common examples include the Pythagorean identities (e.g., sin²(x) + cos²(x) = 1), sum and difference identities, double-angle identities, and half-angle identities.
How to Prove a Trigonometric Identity
Proving a trigonometric identity typically involves starting with one side of the equation (usually the more complex side) and using known identities, algebraic manipulations (like factoring, expanding, finding common denominators), and basic arithmetic to transform it into the other side. The goal is to show that both sides are equivalent for all valid input values.
Key strategies for proving identities:
- Start with the more complicated side: It's usually easier to simplify a complex expression than to complicate a simple one.
- Convert to sines and cosines: Often, expressing all functions in terms of sine and cosine can reveal simplification opportunities.
- Look for algebraic manipulations: Factoring, distributing, combining fractions, or multiplying by conjugates can be useful.
- Use known identities: Substitute one form of an identity for another (e.g., replace
1withsin²(x) + cos²(x)). - Work on both sides independently: Sometimes, simplifying both sides to a common expression is the most efficient path.
Using the Trigonometric Identity Verifier
This calculator provides a numerical check for trigonometric identities. You input an angle and two expressions (representing the left and right sides of a potential identity). The tool then evaluates both expressions at the given angle and compares their results.
Important Note: If the expressions yield approximately equal values for a specific angle, it suggests the identity *might* be true, but it does not constitute a formal proof. An identity must hold true for *all* valid angles. Conversely, if the expressions yield different values, it definitively proves that the proposed statement is *not* an identity.
Input Guidelines:
- Angle Value: Enter the numerical value of the angle.
- Angle Unit: Select whether your angle is in Degrees or Radians.
- Left/Right Side Expression: Enter your trigonometric expressions.
- Use
xas the variable for the angle. - Standard trigonometric functions:
sin(x),cos(x),tan(x),asin(x),acos(x),atan(x). - For powers (e.g., sin²(x)), use
pow(base, exponent). For example,pow(sin(x), 2). - For square roots, use
sqrt(x). - Constants:
PI(for π),E(for Euler's number). - Standard arithmetic operations:
+,-,*,/,().
- Use
Examples of Verification:
Example 1: Pythagorean Identity (sin²(x) + cos²(x) = 1)
- Angle Value: 30
- Angle Unit: Degrees
- Left Side Expression:
pow(sin(x), 2) + pow(cos(x), 2) - Right Side Expression:
1 - Expected Result: Both sides evaluate to 1, indicating they are approximately equal.
Example 2: Double Angle Identity (sin(2x) = 2sin(x)cos(x))
- Angle Value: 60
- Angle Unit: Degrees
- Left Side Expression:
sin(2*x) - Right Side Expression:
2 * sin(x) * cos(x) - Expected Result: Both sides evaluate to approximately 0.866, indicating they are approximately equal.
Example 3: A Non-Identity (sin(x) + cos(x) = 1)
- Angle Value: 45
- Angle Unit: Degrees
- Left Side Expression:
sin(x) + cos(x) - Right Side Expression:
1 - Expected Result: Left side evaluates to approx 1.414, Right side to 1. They are NOT equal, proving it's not an identity.
" + leftExpression + "";
output += "Evaluated Left Side: " + leftResult.toFixed(10) + "";
output += "Right Side Expression: " + rightExpression + "";
output += "Evaluated Right Side: " + rightResult.toFixed(10) + "";
if (Math.abs(leftResult – rightResult) < tolerance) {
output += "Verification Result: Expressions are approximately equal for this angle.";
output += "This suggests the identity might be true, but it is not a formal proof.";
} else {
output += "Verification Result: Expressions are NOT equal for this angle.";
output += "This indicates that the proposed statement is NOT an identity.";
}
resultDiv.innerHTML = output;
}