Arcus Tangent Calculator
Calculate inverse tangent (arctan) values and convert between angles and tangent ratios
Calculate Arcus Tangent
Any real number (domain: ℝ)
Arcus Tangent Result
Quadrant Information
Special Value
arctan(0) = 0°(exact value)
Function: y = arctan(x)
Domain: All real numbers (ℝ)
Range: (-π/2, π/2) or (-90°, 90°)
Example Calculations
Example 1: Finding an Angle
Problem: What angle has a tangent of 1?
Solution: arctan(1) = 45° = π/4 radians
Explanation: The angle whose tangent equals 1 is 45°, which occurs in a 45-45-90 triangle where opposite and adjacent sides are equal.
Example 2: Special Value
Problem: Find arctan(√3)
Solution: arctan(√3) = 60° = π/3 radians
Explanation: In a 30-60-90 triangle, tan(60°) = √3, so arctan(√3) = 60°.
Example 3: Negative Value
Problem: Find arctan(-1)
Solution: arctan(-1) = -45° = -π/4 radians
Explanation: Negative tangent values correspond to angles in the fourth quadrant (or negative angles).
Function Properties
Arctan Function
- • Domain: All real numbers (ℝ)
- • Range: (-π/2, π/2)
- • Odd function: arctan(-x) = -arctan(x)
- • Monotonic: Always increasing
Asymptotes
- • Horizontal: y = π/2 (as x → +∞)
- • Horizontal: y = -π/2 (as x → -∞)
- • No vertical asymptotes
Special Values
Quick Tips
Arctan is the inverse of tangent function
Result is always between -90° and 90°
Also known as tan⁻¹ or inverse tangent
Used to find angles in right triangles
Principal value theorem applies
Understanding Arcus Tangent
What is Arcus Tangent?
Arcus tangent (arctan or tan⁻¹) is the inverse function of tangent. It answers the question: "What angle produces this specific tangent value?" Given a tangent value x, arctan(x) returns the corresponding angle in the range (-π/2, π/2).
Mathematical Definition
If tan(θ) = x, then θ = arctan(x)
where θ ∈ (-π/2, π/2)
Key Properties
- •Domain: All real numbers (-∞, ∞)
- •Range: (-π/2, π/2) or (-90°, 90°)
- •Odd function: arctan(-x) = -arctan(x)
- •Continuous: No breaks or jumps
- •Monotonic: Always increasing
Applications
- • Finding angles in right triangles
- • Converting rectangular to polar coordinates
- • Solving trigonometric equations
- • Navigation and surveying
- • Engineering and physics calculations