Tan Inverse Formula Guide: How to Calculate Arctan in Radians and Degrees
📐 Quick Summary & Formula:
- Definition:
θ = arctan(x) = tan⁻¹(x)wheretan(θ) = x. - Domain & Range: Domain is all real numbers
(-∞, ∞); Range is(-π/2, π/2)or(-90°, 90°). - Radians to Degrees:
Degrees = Radians × (180 / π). - Key Constant:
arctan(1) = 45° = π/4 radians. - Calculate instant angle values with our free Tan Inverse Calculator.
In trigonometry, the tangent function calculates the ratio of the opposite side to the adjacent side of a right-angled triangle. The inverse tangent (commonly written as $\arctan(x)$ or $\tan^{-1}(x)$) performs the exact reverse operation: given a slope or ratio $x$, it returns the corresponding angle $\theta$.
What Is Tan Inverse (Arctan)?
If $\tan(\theta) = x$, then $\arctan(x) = \theta$. For example, because $\tan(45^\circ) = 1$, taking the tan inverse of 1 yields exactly $45^\circ$ (or $\frac{\pi}{4}$ radians).
The Arctan Formula & Mathematical Definition
The inverse tangent function has the following mathematical properties:
- Odd Function Symmetry: $\arctan(-x) = -\arctan(x)$
- Derivative: $\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$
- Taylor Series Expansion (for $|x| \le 1$): $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots$
Converting Between Radians and Degrees
Standard programming languages and scientific calculators compute trigonometric functions in radians. To convert to degrees:
Angle in Degrees = Angle in Radians × (180 ÷ 3.14159265)
Angle in Radians = Angle in Degrees × (3.14159265 ÷ 180)Key Unit Circle Reference Table for Tan Inverse
| Value (x) | Exact Radian Value | Decimal Radians | Angle in Degrees |
|---|---|---|---|
| -√3 (-1.732) | -π/3 | -1.0472 rad | -60.0° |
| -1.000 | -π/4 | -0.7854 rad | -45.0° |
| -1/√3 (-0.577) | -π/6 | -0.5236 rad | -30.0° |
| 0.000 | 0 | 0.0000 rad | 0.0° |
| 1/√3 (0.577) | π/6 | 0.5236 rad | 30.0° |
| 1.000 | π/4 | 0.7854 rad | 45.0° |
| √3 (1.732) | π/3 | 1.0472 rad | 60.0° |
Understanding atan(x) vs. atan2(y, x) in Programming
In software development and robotics, computing angles using Math.atan(y / x) introduces ambiguity because division loses the signs of $x$ and $y$. The Math.atan2(y, x) function resolves this by inspecting both signs to return angles across all four quadrants ($-180^\circ$ to $+180^\circ$).
Real-World Engineering, Physics & Construction Applications
- Roof Pitch & Slope Inclines: Converting rise-over-run slope measurements into pitch angles in degrees (e.g. 4/12 pitch has angle $\arctan(4/12) = 18.43^\circ$).
- Wheelchair Ramp Compliance (ADA): Ensuring ramp slope does not exceed $1:12$ (an incline angle of $\arctan(1/12) = 4.76^\circ$).
- Electrical AC Power Factor: Computing the phase angle $\theta = \arctan(X / R)$ between reactance $X$ and resistance $R$.
Try computing custom angles with our free Tan Inverse Calculator.
Dr. Vance is a research mathematician and psychometric modeler specializing in standardized test scoring distributions, inverse trigonometry, and algebraic weighted averages.
