Inverse Tangent Calculator
Understanding the Inverse Tangent (Arctan)
The inverse tangent, often denoted as arctan 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. In simpler terms, if you know the tangent of an angle, the inverse tangent function tells you what that angle is.
How it Works
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 to θ to the length of the side adjacent to θ:
tan(θ) = Opposite / Adjacent
The inverse tangent function reverses this operation. If you have the ratio (Opposite / Adjacent), you can find the angle θ using:
θ = arctan(Opposite / Adjacent)
This calculator takes the lengths of the opposite and adjacent sides as input and provides the angle in degrees.
Applications of Inverse Tangent
- Finding Angles in Geometry: When you know the side lengths of a right triangle, you can use arctan to determine its acute angles.
- Vector Analysis: In physics and engineering, arctan is used to find the direction (angle) of a vector given its x and y components. For example, if a vector has components (x, y), its angle with the positive x-axis can be found using arctan(y/x).
- Coordinate Transformations: Converting Cartesian coordinates (x, y) to polar coordinates (r, θ) often involves the use of arctan to find θ.
- Computer Graphics: Used in various algorithms for rotations and transformations.
Important Considerations
The standard `atan()` function in most programming languages (and this calculator) returns an angle in the range of -90° to +90° (or -π/2 to +π/2 radians). This is because the tangent function has a period of π (180°), and its inverse is typically defined within this principal value range.
If you need to determine an angle in all four quadrants (0° to 360°), you might need to use a function like `atan2(y, x)` (which takes both the opposite and adjacent components separately) and consider the signs of both inputs. However, for this basic calculator, we assume a standard right-triangle context or a principal value.
Example Calculation
Let's say you have a right triangle where the side opposite to the angle is 3 units long, and the side adjacent to the angle is 4 units long.
- Opposite Side Length: 3
- Adjacent Side Length: 4
- Ratio: 3 / 4 = 0.75
- Inverse Tangent: arctan(0.75) ≈ 36.87 degrees
Using the calculator above, input '3' for Opposite Side Length and '4' for Adjacent Side Length, then click 'Calculate' to verify this result.