Arctangent Calculator
Enter a numerical value to calculate its arctangent (inverse tangent) in both radians and degrees.
Understanding the Arctangent Function
The arctangent function, often denoted as atan(x) or tan-1(x), is the inverse operation of the tangent function. While the tangent function takes an angle and returns the ratio of the opposite side to the adjacent side in a right-angled triangle, the arctangent function takes this ratio (a numerical value) and returns the corresponding angle.
What Does Arctangent Calculate?
In essence, if you know the tangent of an angle, the arctangent function helps you find that angle. For example, if tan(θ) = x, then θ = atan(x). The result of the arctangent function is typically given in radians, but it can also be converted to degrees for easier interpretation.
- Radians: The standard unit for angles in mathematics, especially in calculus and physics. A full circle is 2π radians.
- Degrees: A more commonly understood unit for angles, where a full circle is 360 degrees.
Applications of Arctangent
The arctangent function has numerous applications across various fields:
- Geometry and Trigonometry: It's fundamental for finding unknown angles in right-angled triangles when the lengths of the opposite and adjacent sides are known. For instance, if you have a ramp and know its height and horizontal length, you can use arctan to find its angle of inclination.
- Physics and Engineering:
- Vector Analysis: Used to determine the direction or phase angle of a vector given its components (e.g., finding the angle of a force vector or velocity vector).
- Electrical Engineering: Essential for calculating phase angles in AC circuits, especially when dealing with impedance (resistance, capacitance, and inductance).
- Robotics and Navigation: Used in algorithms for determining orientation and direction.
- Computer Graphics: Employed in 2D and 3D transformations to rotate objects or align cameras.
- Calculus: The derivative of arctan(x) is 1/(1+x2), making it important in integration techniques.
Examples of Arctangent Values
Let's look at some common arctangent values:
- atan(0): The angle whose tangent is 0 is 0 radians (0 degrees).
- atan(1): The angle whose tangent is 1 is π/4 radians (45 degrees). This occurs in an isosceles right triangle where opposite and adjacent sides are equal.
- atan(√3): The angle whose tangent is √3 is π/3 radians (60 degrees).
- atan(1/√3): The angle whose tangent is 1/√3 is π/6 radians (30 degrees).
- atan(-1): The angle whose tangent is -1 is -π/4 radians (-45 degrees). The arctangent function typically returns values in the range of (-π/2, π/2) or (-90°, 90°).
Using the calculator above, you can input any real number and instantly find its arctangent in both radians and degrees, helping you understand these concepts better.