Tan Inverse Formula Guide: How to Calculate Arctan in Radians and Degrees

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

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.