Trigonometry is a fundamental branch of mathematics that studies the relationships between angles and sides of triangles. The three most basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are defined for angles and relate them to ratios of sides in a right-angled triangle, or to coordinates on the unit circle. They are indispensable tools in fields like physics, engineering, navigation, computer graphics, and many areas of scientific research.
The Right-Angled Triangle Definition
Consider a right-angled triangle with one angle denoted as $\theta$. The sides relative to this angle are:
Opposite: The side directly across from angle $\theta$.
Adjacent: The side next to angle $\theta$ (that is not the hypotenuse).
Hypotenuse: The longest side, opposite the right angle.
Using these definitions, the trigonometric functions are defined as ratios:
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent
The Unit Circle Definition
Trigonometric functions can also be understood using the unit circle (a circle with radius 1 centered at the origin of a coordinate plane). For any angle $\theta$ measured counterclockwise from the positive x-axis, the point where the terminal side of the angle intersects the unit circle has coordinates (x, y).
sin(θ) is the y-coordinate of the point.
cos(θ) is the x-coordinate of the point.
tan(θ) is the ratio y/x.
This definition extends the trigonometric functions to all real numbers, including angles greater than 90 degrees or less than 0 degrees.
Why Use a Trigonometric Calculator?
While the concepts are simple, manual calculation of trigonometric values, especially for angles that are not simple fractions of $\pi$ radians or 90 degrees, can be complex and time-consuming. Calculators simplify this by:
Providing quick and accurate values for sin, cos, and tan.
Handling conversions between degrees and radians, which are common units for angle measurement.
Allowing users to explore the behavior of these functions for various inputs.
Common Use Cases:
Physics: Analyzing projectile motion, wave phenomena (sound, light, AC circuits), oscillations, and forces.
Engineering: Designing structures, electrical circuits, signal processing, and mechanical systems.
Navigation: Calculating positions and bearings.
Geometry: Solving triangle problems and working with periodic functions.
Computer Graphics: Implementing rotations and transformations.
This calculator allows you to input an angle (in degrees or radians) and select the trigonometric function (sine, cosine, or tangent) to get an immediate result.
function calculateTrig() {
var angleValue = parseFloat(document.getElementById("angleValue").value);
var angleUnit = document.getElementById("angleUnit").value;
var functionType = document.getElementById("functionType").value;
var resultDisplay = document.getElementById("result-value");
if (isNaN(angleValue)) {
resultDisplay.textContent = "Invalid Input";
return;
}
var angleInRadians;
if (angleUnit === "degrees") {
angleInRadians = angleValue * Math.PI / 180;
} else {
angleInRadians = angleValue;
}
var result = 0;
if (functionType === "sin") {
result = Math.sin(angleInRadians);
} else if (functionType === "cos") {
result = Math.cos(angleInRadians);
} else if (functionType === "tan") {
// Special handling for tan(90 degrees) or tan(pi/2 radians) which is undefined.
// Due to floating point precision, we check for values very close to pi/2 + n*pi
var angleInDegrees = angleValue;
if (angleUnit === "radians") {
angleInDegrees = angleValue * 180 / Math.PI;
}
// Check if angle is close to 90, 270, -90, etc. degrees
// (n * 180 + 90) degrees
var remainder = Math.abs(angleInDegrees) % 180;
if (Math.abs(remainder – 90) < 1e-10) { // Use a small tolerance for floating point comparisons
resultDisplay.textContent = "Undefined";
return;
}
result = Math.tan(angleInRadians);
}
// Display result with a reasonable number of decimal places
resultDisplay.textContent = result.toFixed(10).replace(/\.?0+$/, ''); // Remove trailing zeros
}