Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. At its core, it deals with the six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are fundamental in fields like physics, engineering, computer graphics, navigation, and many areas of pure mathematics.
The Basics: Unit Circle and Definitions
The trigonometric functions can be understood by considering a unit circle (a circle with a radius of 1 centered at the origin of a Cartesian coordinate system) and an angle θ originating from the positive x-axis and rotating counterclockwise. For any point (x, y) on the unit circle corresponding to this angle:
sin(θ) = y (the y-coordinate)
cos(θ) = x (the x-coordinate)
tan(θ) = sin(θ) / cos(θ) = y / x (provided cos(θ) ≠ 0)
The other three functions are the reciprocals of these:
csc(θ) = 1 / sin(θ) = 1 / y (provided sin(θ) ≠ 0)
sec(θ) = 1 / cos(θ) = 1 / x (provided cos(θ) ≠ 0)
cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ) = x / y (provided sin(θ) ≠ 0)
Angle Units: Degrees vs. Radians
Angles can be measured in two primary units: degrees and radians.
Degrees: A full circle is 360 degrees. A right angle is 90 degrees.
Radians: A full circle is 2π radians. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. The conversion is: 180 degrees = π radians.
Most programming languages and calculators use radians for their internal trigonometric functions, so it's crucial to ensure your input angle is in the correct unit or converted appropriately. This calculator allows you to specify the unit.
Engineering: Designing structures, analyzing electrical circuits (AC), signal processing, and robotics.
Navigation: Calculating distances and bearings, especially in GPS systems and surveying.
Computer Graphics: Rotating objects, creating animations, and rendering 3D scenes.
Mathematics: Solving triangles, calculus (integrals and derivatives of trig functions), and complex number analysis.
This calculator provides a quick and easy way to compute these fundamental values for any given angle and selected function.
function calculateTrig() {
var angleValueInput = document.getElementById("angleValue");
var angleUnitSelect = document.getElementById("angleUnit");
var functionTypeSelect = document.getElementById("functionType");
var resultDiv = document.getElementById("result");
var angleValue = parseFloat(angleValueInput.value);
var angleUnit = angleUnitSelect.value;
var functionType = functionTypeSelect.value;
if (isNaN(angleValue)) {
resultDiv.innerHTML = "Error: Please enter a valid number for the angle.";
return;
}
var angleInRadians = angleValue;
if (angleUnit === "degrees") {
angleInRadians = angleValue * (Math.PI / 180);
}
var result;
var functionName = ";
try {
switch (functionType) {
case "sin":
result = Math.sin(angleInRadians);
functionName = 'sin';
break;
case "cos":
result = Math.cos(angleInRadians);
functionName = 'cos';
break;
case "tan":
var cosVal = Math.cos(angleInRadians);
if (cosVal === 0) throw new Error("Tangent is undefined when cosine is zero (angle is π/2 + nπ).");
result = Math.tan(angleInRadians);
functionName = 'tan';
break;
case "csc":
var sinVal = Math.sin(angleInRadians);
if (sinVal === 0) throw new Error("Cosecant is undefined when sine is zero (angle is nπ).");
result = 1 / sinVal;
functionName = 'csc';
break;
case "sec":
var cosVal = Math.cos(angleInRadians);
if (cosVal === 0) throw new Error("Secant is undefined when cosine is zero (angle is π/2 + nπ).");
result = 1 / cosVal;
functionName = 'sec';
break;
case "cot":
var sinVal = Math.sin(angleInRadians);
if (sinVal === 0) throw new Error("Cotangent is undefined when sine is zero (angle is nπ).");
result = 1 / Math.tan(angleInRadians);
functionName = 'cot';
break;
default:
throw new Error("Unknown function type.");
}
// Handle potential floating point inaccuracies for common values like 0, 1, -1
if (Math.abs(result) < 1e-10) {
result = 0;
} else if (Math.abs(result – 1) < 1e-10) {
result = 1;
} else if (Math.abs(result + 1) < 1e-10) {
result = -1;
}
resultDiv.innerHTML = functionName + "(" + angleValue + "°/" + (angleValue * (Math.PI / 180)).toFixed(4) + " rad) = " + result.toFixed(6) + "";
} catch (error) {
resultDiv.innerHTML = "Error: " + error.message + "";
}
}