Calculator with Inverse Tangent

Inverse Tangent (Arctan) Calculator

Result:

Angle (Radians): –

Angle (Degrees): –

function calculateInverseTangent() { var oppositeSide = parseFloat(document.getElementById('oppositeSide').value); var adjacentSide = parseFloat(document.getElementById('adjacentSide').value); var angleRadiansElement = document.getElementById('angleRadians'); var angleDegreesElement = document.getElementById('angleDegrees'); var resultDiv = document.getElementById('inverseTangentResult'); if (isNaN(oppositeSide) || isNaN(adjacentSide)) { angleRadiansElement.innerHTML = 'Angle (Radians): Please enter valid numbers for both sides.'; angleDegreesElement.innerHTML = 'Angle (Degrees): -'; resultDiv.style.backgroundColor = '#ffe0e0'; // Light red for error return; } if (adjacentSide === 0) { if (oppositeSide === 0) { angleRadiansElement.innerHTML = 'Angle (Radians): Undefined (both sides are zero)'; angleDegreesElement.innerHTML = 'Angle (Degrees): -'; resultDiv.style.backgroundColor = '#ffe0e0'; return; } else if (oppositeSide > 0) { // For positive opposite and zero adjacent, angle is 90 degrees (PI/2) var angleRad = Math.PI / 2; var angleDeg = 90; angleRadiansElement.innerHTML = 'Angle (Radians): ' + angleRad.toFixed(4) + ' rad'; angleDegreesElement.innerHTML = 'Angle (Degrees): ' + angleDeg.toFixed(2) + ' °'; resultDiv.style.backgroundColor = '#e9ecef'; return; } else { // oppositeSide < 0 // For negative opposite and zero adjacent, angle is -90 degrees (-PI/2) var angleRad = -Math.PI / 2; var angleDeg = -90; angleRadiansElement.innerHTML = 'Angle (Radians): ' + angleRad.toFixed(4) + ' rad'; angleDegreesElement.innerHTML = 'Angle (Degrees): ' + angleDeg.toFixed(2) + ' °'; resultDiv.style.backgroundColor = '#e9ecef'; return; } } var ratio = oppositeSide / adjacentSide; var angleRad = Math.atan(ratio); var angleDeg = angleRad * (180 / Math.PI); angleRadiansElement.innerHTML = 'Angle (Radians): ' + angleRad.toFixed(4) + ' rad'; angleDegreesElement.innerHTML = 'Angle (Degrees): ' + angleDeg.toFixed(2) + ' °'; resultDiv.style.backgroundColor = '#e9ecef'; // Reset background on success }

Understanding the Inverse Tangent (Arctan)

The inverse tangent, often denoted as arctan, atan, or tan^-1, is a fundamental trigonometric function used to find the angle when you know the ratio of the opposite side to the adjacent side in a right-angled triangle. It's the inverse operation of the tangent function.

What is Tangent?

In a right-angled triangle, the tangent of an angle (let's call it θ) is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically:

tan(θ) = Opposite / Adjacent

What is Inverse Tangent (Arctan)?

When you know the lengths of the opposite and adjacent sides, but you need to find the angle θ itself, you use the inverse tangent function. It "undoes" the tangent operation:

θ = arctan(Opposite / Adjacent)

The result of the arctan function is an angle, which can be expressed in either radians or degrees.

When is Arctan Used?

The inverse tangent is incredibly useful in various fields:

• Geometry and Trigonometry: Finding unknown angles in right-angled triangles when side lengths are known.
• Physics: Calculating the angle of a resultant vector, determining the angle of inclination, or analyzing projectile motion.
• Engineering: Used in structural analysis, robotics, and computer graphics to determine orientations and angles.
• Computer Science: Essential in game development for character movement and camera control, and in algorithms involving spatial relationships.

Radians vs. Degrees

Angles can be measured in two primary units:

• Degrees (°): A full circle is 360 degrees. This is commonly used in everyday applications and many engineering contexts.
• Radians (rad): A full circle is 2π radians. Radians are often preferred in higher-level mathematics, physics, and calculus because they simplify many formulas.

This calculator provides the angle in both units for your convenience.

How to Use the Inverse Tangent Calculator

Our calculator simplifies the process of finding an angle using the inverse tangent:

1. Enter Opposite Side Length: Input the numerical value for the side opposite the angle you want to find.
2. Enter Adjacent Side Length: Input the numerical value for the side adjacent to the angle.
3. Click "Calculate Angle": The calculator will instantly display the angle in both radians and degrees.

Examples:

• Example 1: Isosceles Right Triangle
  If the Opposite Side Length = 1 and the Adjacent Side Length = 1:
  Ratio = 1 / 1 = 1
  θ = arctan(1) = 0.7854 radians (or 45 degrees).
• Example 2: Common Right Triangle
  If the Opposite Side Length = 1.732 (approx. √3) and the Adjacent Side Length = 1:
  Ratio = 1.732 / 1 = 1.732
  θ = arctan(1.732) = 1.0472 radians (or 60 degrees).
• Example 3: Horizontal Line
  If the Opposite Side Length = 0 and the Adjacent Side Length = 5:
  Ratio = 0 / 5 = 0
  θ = arctan(0) = 0 radians (or 0 degrees).
• Example 4: Vertical Line (Special Case)
  If the Opposite Side Length = 5 and the Adjacent Side Length = 0:
  Ratio = 5 / 0 (undefined, approaches infinity)
  θ = arctan(∞) = 1.5708 radians (or 90 degrees). The calculator handles this edge case correctly.

Use this tool to quickly and accurately determine angles for your mathematical, scientific, or engineering problems.