Calculate Arctangent

Arctangent Calculator

Understanding the Arctangent Function

The arctangent function, often denoted as atan(x), arctan(x), or tan⁻¹(x), is the inverse operation of the tangent function. In simple terms, if you know the tangent of an angle, the arctangent function helps you find that angle. It answers the question: "What angle has a tangent equal to this given value?"

What Does Arctangent Calculate?

In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side (Tangent = Opposite / Adjacent). The arctangent function takes this ratio as its input and returns the angle itself. For example, if the ratio of the opposite side to the adjacent side is 1, the arctangent of 1 is 45 degrees (or π/4 radians).

Domain and Range

• Domain: The input value 'x' for the arctangent function can be any real number, from negative infinity to positive infinity.
• Range: The output angle of atan(x) is typically restricted to a range of -π/2 to π/2 radians (or -90° to 90°). This restriction ensures that for every input 'x', there is a unique output angle.

Radians vs. Degrees

Angles can be measured in two primary units: radians and degrees. Our calculator provides results in both:

• Degrees: A full circle is 360 degrees. This is the most commonly understood unit for angles in everyday contexts.
• Radians: A full circle is 2π radians. Radians are often preferred in mathematics, physics, and engineering because they simplify many formulas, especially in calculus. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

The conversion between them is: Degrees = Radians × (180 / π) and Radians = Degrees × (π / 180).

Applications of Arctangent

The arctangent function is incredibly useful in various fields:

• Trigonometry: Finding unknown angles in right-angled triangles when the lengths of the opposite and adjacent sides are known.
• Physics and Engineering: Calculating phase angles in AC circuits, determining the angle of a resultant vector, or finding the angle of inclination.
• Computer Graphics and Game Development: Used to orient objects, calculate trajectories, and determine viewing angles.
• Robotics: Essential for inverse kinematics, where you need to find the joint angles required to reach a specific position.

The atan2(y, x) Function (Beyond Simple Arctangent)

While atan(x) is useful, it has a limitation: its output is restricted to -90° to 90°. This means it cannot distinguish between angles in the first and third quadrants (where tangent is positive) or the second and fourth quadrants (where tangent is negative). For example, atan(1) is 45°, but an angle of 225° also has a tangent of 1.

To overcome this, many programming languages and mathematical libraries offer an atan2(y, x) function. This function takes two arguments, a 'y' coordinate and an 'x' coordinate, and returns the angle in the full range of -π to π radians (-180° to 180°), correctly identifying the quadrant of the angle. Our calculator focuses on the simpler atan(x), but it's important to be aware of atan2 for more complex angular calculations.

How to Use This Calculator

Simply enter the numerical value for which you want to find the arctangent into the "Value (x)" field. Click "Calculate Arctangent," and the calculator will instantly display the corresponding angle in both radians and degrees.

Examples:

• If x = 1:
  • Arctangent (Radians): 0.785398 rad (which is π/4)
  • Arctangent (Degrees): 45°
• If x = 0:
  • Arctangent (Radians): 0 rad
  • Arctangent (Degrees): 0°
• If x = -1:
  • Arctangent (Radians): -0.785398 rad (which is -π/4)
  • Arctangent (Degrees): -45°
• If x = 1.73205 (approx. tan(60°)):
  • Arctangent (Radians): 1.047198 rad (which is π/3)
  • Arctangent (Degrees): 60°