Trigonometric Identity Calculator
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Trigonometric Identity Calculator
Result will be displayed here.
Understanding Trigonometric Identities
Trigonometric identities are fundamental equations that hold true for all valid input values. They are crucial in various fields, including mathematics, physics, engineering, and signal processing, as they allow us to simplify complex expressions, solve trigonometric equations, and prove other identities.
This calculator focuses on evaluating the basic trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) for a given angle. While not an "identity solver" in the sense of manipulating equations, it provides the fundamental values upon which many identities are based. Knowing these values is the first step in working with trigonometric identities.
Key Trigonometric Functions:
- Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle.
- Cosine (cos): The ratio of the length of the adjacent side to an angle to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. It's also equal to sin(θ) / cos(θ).
- Cosecant (csc): The reciprocal of sine (1 / sin(θ)).
- Secant (sec): The reciprocal of cosine (1 / cos(θ)).
- Cotangent (cot): The reciprocal of tangent (1 / tan(θ)). It's also equal to cos(θ) / sin(θ).
Common Trigonometric Identities:
While this calculator directly computes function values, understanding identities is key. Some common ones include:
- Pythagorean Identities:
sin²(θ) + cos²(θ) = 11 + tan²(θ) = sec²(θ)1 + cot²(θ) = csc²(θ)
- Reciprocal Identities:
csc(θ) = 1 / sin(θ)sec(θ) = 1 / cos(θ)cot(θ) = 1 / tan(θ)
- Quotient Identities:
tan(θ) = sin(θ) / cos(θ)cot(θ) = cos(θ) / sin(θ)
- Sum and Difference Identities, Double Angle Identities, Half Angle Identities, Product-to-Sum Identities, Sum-to-Product Identities, etc.
How to Use This Calculator:
Select the trigonometric function you want to evaluate. Enter the angle value and choose whether the angle is in degrees or radians. The calculator will then display the computed value of the selected trigonometric function for that angle. This is useful for checking specific values when working with trigonometric equations or proofs.
Example Calculation:
Let's calculate the Sine of 30 degrees.
- Function: Sine (sin)
- Angle: 30
- Unit: Degrees
- Expected Result: 0.5
Another example: Calculate the Tangent of π/4 radians.
- Function: Tangent (tan)
- Angle: 0.785398 (approximately π/4)
- Unit: Radians
- Expected Result: 1 (approximately)
function convertToRadians(angle, unit) {
if (unit === 'deg') {
return angle * Math.PI / 180;
} else if (unit === 'rad') {
return angle;
}
return 0; // Default or error case
}
function updateAngleUnits() {
var functionType = document.getElementById("functionType").value;
var angleLabel = document.getElementById("angleLabel");
var unitTypeSelect = document.getElementById("unitType");
if (functionType === 'sin' || functionType === 'cos' || functionType === 'tan' ||
functionType === 'csc' || functionType === 'sec' || functionType === 'cot') {
angleLabel.textContent = "Angle:";
unitTypeSelect.disabled = false;
// Optionally, you could try to pre-select degrees as a common default
// unitTypeSelect.value = 'deg';
}
}
function calculateTrigIdentity() {
var functionType = document.getElementById("functionType").value;
var angleInput = document.getElementById("angle");
var unitType = document.getElementById("unitType").value;
var resultDiv = document.getElementById("result");
var angleValue = parseFloat(angleInput.value);
if (isNaN(angleValue)) {
resultDiv.textContent = "Please enter a valid angle.";
return;
}
var angleInRadians = convertToRadians(angleValue, unitType);
var result = NaN;
var displayResult = "";
try {
switch (functionType) {
case 'sin':
result = Math.sin(angleInRadians);
break;
case 'cos':
result = Math.cos(angleInRadians);
break;
case 'tan':
// Handle tan(pi/2 + n*pi) which is undefined
if (Math.abs(Math.cos(angleInRadians)) < 1e-10) {
result = Infinity; // Represents undefined
} else {
result = Math.tan(angleInRadians);
}
break;
case 'csc':
var sinVal = Math.sin(angleInRadians);
if (Math.abs(sinVal) < 1e-10) {
result = Infinity; // Represents undefined
} else {
result = 1 / sinVal;
}
break;
case 'sec':
var cosVal = Math.cos(angleInRadians);
if (Math.abs(cosVal) < 1e-10) {
result = Infinity; // Represents undefined
} else {
result = 1 / cosVal;
}
break;
case 'cot':
// Handle cot(n*pi) which is undefined
if (Math.abs(Math.sin(angleInRadians)) < 1e-10) {
result = Infinity; // Represents undefined
} else {
result = 1 / Math.tan(angleInRadians);
}
break;
default:
resultDiv.textContent = "Invalid function selected.";
return;
}
// Format the result for better readability, especially for close-to-zero values or undefined cases
if (result === Infinity || result === -Infinity) {
displayResult = "Undefined";
} else if (Math.abs(result) < 1e-10) { // Handle values very close to zero
displayResult = "0";
}
else {
displayResult = result.toFixed(10); // Display with a reasonable precision
}
resultDiv.textContent = `${functionType}(${angleValue}°) = ${displayResult}`; // Assuming degrees for display for simplicity, or adjust based on selected unit
} catch (error) {
resultDiv.textContent = "An error occurred.";
console.error("Calculation error:", error);
}
}
// Initial call to set up the angle unit label correctly on load
updateAngleUnits();