The tangent function, denoted as tan(θ), is a fundamental trigonometric function that plays a crucial role in mathematics, physics, engineering, and many other scientific disciplines. It is defined for a right-angled triangle as the ratio of the length of the side opposite to an angle to the length of the side adjacent to that angle.
Mathematical Definition
In a right-angled triangle:
Opposite: The side directly across from the angle θ.
Adjacent: The side next to the angle θ (and not the hypotenuse).
Therefore, the tangent is defined as:
tan(θ) = Opposite / Adjacent
Beyond right-angled triangles, the tangent function can be understood using the unit circle. For an angle θ measured counterclockwise from the positive x-axis, the tangent is the y-coordinate divided by the x-coordinate of the point where the terminal side of the angle intersects the unit circle. It can also be defined as the ratio of sine to cosine:
tan(θ) = sin(θ) / cos(θ)
Units of Measurement
Angles can be measured in two primary units:
Degrees: A full circle is 360 degrees (°).
Radians: A full circle is 2π radians. Radians are often preferred in calculus and higher mathematics due to their direct relationship with arc length and angles in geometry.
This calculator allows you to input your angle in either degrees or radians, ensuring flexibility for various applications.
Key Properties and Use Cases
Slope: The tangent of an angle represents the slope of a line that makes that angle with the positive x-axis. For example, if a line makes an angle of 45° with the x-axis, its slope is tan(45°) = 1.
Physics: Used in projectile motion, wave analysis, and vector decomposition. For instance, it can help calculate the trajectory of a projectile or the resultant force.
Engineering: Essential in surveying, navigation, structural design, and electrical engineering to calculate distances, angles, and forces.
Trigonometry and Calculus: A core function used in solving triangles, understanding periodic functions, and deriving integrals and derivatives.
Asymptotes: The tangent function has vertical asymptotes at angles where the cosine is zero (e.g., 90°, 270°, and their equivalents in radians, π/2, 3π/2).
Example Calculation
Let's calculate the tangent of 45 degrees:
Input Angle Value: 45
Select Unit: Degrees
The calculator will compute tan(45°). Mathematically, in a right isosceles triangle, the opposite and adjacent sides are equal, so the ratio is 1. The result is 1.
Consider an angle of π/4 radians (which is equivalent to 45 degrees):
Input Angle Value: 0.785398 (approx. π/4)
Select Unit: Radians
The calculator will compute tan(π/4), yielding a result very close to 1.
The tangent function is undefined at 90 degrees (or π/2 radians). Attempting to calculate it will result in an error or infinity, reflecting the vertical slope of the line at that angle.
function calculateTangent() {
var angleValue = parseFloat(document.getElementById("angleValue").value);
var angleUnit = document.getElementById("angleUnit").value;
var tangentResultElement = document.getElementById("tangentResult");
var resultContainer = document.getElementById("result-container");
if (isNaN(angleValue)) {
tangentResultElement.textContent = "Invalid input";
tangentResultElement.style.color = "red";
resultContainer.style.display = "block";
return;
}
var angleInRadians;
if (angleUnit === "degrees") {
// Convert degrees to radians
angleInRadians = angleValue * (Math.PI / 180);
} else {
// Input is already in radians
angleInRadians = angleValue;
}
// Check for angles where tangent is undefined (multiples of PI/2 plus PI/2)
// We use a small tolerance for floating point comparisons
var tolerance = 1e-10;
var remainder = angleInRadians % Math.PI;
if (Math.abs(remainder – Math.PI / 2) < tolerance || Math.abs(remainder + Math.PI / 2) < tolerance) {
tangentResultElement.textContent = "Undefined (approaching infinity)";
tangentResultElement.style.color = "orange";
} else {
var tangentValue = Math.tan(angleInRadians);
tangentResultElement.textContent = tangentValue.toFixed(6); // Display with 6 decimal places
tangentResultElement.style.color = "#28a745"; // Success green
}
resultContainer.style.display = "block";
}