Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles, particularly right-angled triangles. It is fundamental to many fields including physics, engineering, navigation, surveying, and computer graphics.
Key Trigonometric Functions
The three primary trigonometric functions are Sine (sin), Cosine (cos), and Tangent (tan). For a right-angled triangle with an angle θ, and sides labeled opposite, adjacent, and hypotenuse:
Sine (sin θ): The ratio of the length of the side opposite the angle to the length of the hypotenuse. sin θ = Opposite / Hypotenuse
Cosine (cos θ): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos θ = Adjacent / Hypotenuse
Tangent (tan θ): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. tan θ = Opposite / Adjacent
Inverse Trigonometric Functions
The inverse trigonometric functions (also known as arcs functions) are used to find the angle when you know the ratio of sides. They are denoted as arcsin (or sin⁻¹), arccos (or cos⁻¹), and arctan (or tan⁻¹).
ArcSine (asin y): Returns the angle whose sine is y.
ArcCosine (acos y): Returns the angle whose sine is y.
ArcTangent (atan y): Returns the angle whose tangent is y.
The output of inverse trigonometric functions is typically given in radians or degrees.
Angle Units: Degrees vs. Radians
Trigonometric calculations often require specifying whether the angle is in degrees or radians:
Degrees: A full circle is 360 degrees (°).
Radians: A full circle is 2π radians. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
180 degrees = π radians
Calculator Usage
This calculator allows you to:
Calculate the sine, cosine, or tangent of an angle (given in degrees or radians).
Calculate the angle (in degrees or radians) whose sine, cosine, or tangent is a given ratio.
Simply input the numerical value, select the desired operation, and specify the unit of the angle if applicable. The calculator will then display the result.
Applications of Trigonometry
Trigonometry is indispensable in fields such as:
Physics: Analyzing wave motion, oscillations, and forces.
Engineering: Designing structures, circuits, and mechanical systems.
Navigation: Determining positions and courses for ships and aircraft.
Astronomy: Calculating distances to stars and planets.
Computer Graphics: Rendering 3D models and animations.
function calculateTrig() {
var valueInput = document.getElementById('value');
var operation = document.getElementById('operation').value;
var angleUnit = document.getElementById('angle_unit').value;
var resultDisplay = document.getElementById('result');
var resultContainer = document.getElementById('result-container');
var errorMessage = document.getElementById('error-message');
errorMessage.style.display = 'none';
resultContainer.style.display = 'none';
var value = parseFloat(valueInput.value);
if (isNaN(value)) {
errorMessage.textContent = 'Please enter a valid number for the angle or ratio.';
errorMessage.style.display = 'block';
return;
}
var result;
if (angleUnit === 'degrees') {
value = value * (Math.PI / 180); // Convert degrees to radians for internal calculations
}
switch (operation) {
case 'sin':
result = Math.sin(value);
break;
case 'cos':
result = Math.cos(value);
break;
case 'tan':
// Avoid division by zero for tan(90°), tan(270°), etc.
if (Math.abs(Math.cos(value)) < 1e-10) { // Check if close to zero
errorMessage.textContent = 'Tangent is undefined for this angle (approaching 90°, 270°, etc.).';
errorMessage.style.display = 'block';
return;
}
result = Math.tan(value);
break;
case 'asin':
// Domain for asin is [-1, 1]
if (value 1) {
errorMessage.textContent = 'ArcSine input must be between -1 and 1.';
errorMessage.style.display = 'block';
return;
}
result = Math.asin(value);
break;
case 'acos':
// Domain for acos is [-1, 1]
if (value 1) {
errorMessage.textContent = 'ArcCosine input must be between -1 and 1.';
errorMessage.style.display = 'block';
return;
}
result = Math.acos(value);
break;
case 'atan':
result = Math.atan(value);
break;
default:
errorMessage.textContent = 'Invalid operation selected.';
errorMessage.style.display = 'block';
return;
}
// Convert result back to degrees if the original input was degrees and the operation is an inverse one
if (angleUnit === 'degrees' && (operation === 'asin' || operation === 'acos' || operation === 'atan')) {
result = result * (180 / Math.PI);
}
// Display result with appropriate formatting
var formattedResult = result.toFixed(8); // Limit decimal places for clarity
resultDisplay.textContent = formattedResult;
resultContainer.style.display = 'block';
}