Example 1 — Special ratios
\(\arctan(1)=\tfrac{\pi}{4}=45^\circ\), \(\arctan(\sqrt{3})=\tfrac{\pi}{3}=60^\circ\), \(\arctan(-1)=-\tfrac{\pi}{4}=-45^\circ\).
Inverse Tangent Calculator
Compute \(\theta=\arctan(x)\) (principal value). Enter the tangent value and press Calculate.
You can type numbers or expressions (e.g., sqrt(3), 1/3). Output shows radians and degrees.
Equation Preview
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Helping notes
\( \text{Definition: } \theta=\arctan(x) \text{ is the angle with } \tan(\theta)=x,\; \theta\in(-\tfrac{\pi}{2},\,\tfrac{\pi}{2}). \)
\( \text{Common values: } \arctan(0)=0,\; \arctan(1)=\tfrac{\pi}{4},\; \arctan(\sqrt{3})=\tfrac{\pi}{3},\; \arctan(1/\sqrt{3})=\tfrac{\pi}{6}. \)
\( \text{Tip: } x=\tfrac{\text{opposite}}{\text{adjacent}} \text{ in a right triangle; signs determine } \theta\in(-\tfrac{\pi}{2},\,\tfrac{\pi}{2}). \)
Results
Angle (principal value)
Exact Form (when recognizable)
Check
What is an Inverse Tangent Calculator?
An Inverse Tangent Calculator evaluates \(\arctan(x)\) (also written \(\tan^{-1}(x)\)) to convert a slope or ratio into an angle. The principal-value function maps real inputs to angles in the open interval \((-\tfrac{\pi}{2},\,\tfrac{\pi}{2})\). In right-triangle terms, \(x=\tfrac{\text{opposite}}{\text{adjacent}}\) and \(\theta=\arctan(x)\). For coordinate geometry, the two-argument form \(\operatorname{atan2}(y,x)\) returns the correct quadrant of the direction angle from the origin to \((x,y)\). This tool handles exact values when recognizable (e.g., \(\arctan(1)=\tfrac{\pi}{4}\)) and produces high-precision decimals otherwise. It can switch units (degrees ↔ radians), apply identities to simplify expressions, and show derivations clearly.
About the Inverse Tangent Calculator
The calculator detects domain issues, explains principal values, and addresses common quadrant pitfalls by recommending \(\operatorname{atan2}\) for vectors. It displays unit conversions, exact symbolic outcomes for special angles, and numerical approximations with chosen precision. Advanced options include combining angles using addition formulas, differentiating \(\arctan(x)\) for calculus contexts, and interpreting negative or large inputs. For modeling, it interprets slopes from lines or regressions: given a slope \(m\), the inclination is \(\arctan(m)\) (or \(\operatorname{atan2}(\Delta y,\Delta x)\) for segment endpoints). The output includes step-by-step algebra and a concise summary of angle, unit, quadrant, and any simplifications used.
How to Use this Inverse Tangent Calculator
- Enter a number \(x\) (ratio/slope), or a vector \((x,y)\) to use \(\operatorname{atan2}(y,x)\).
- Select output units (radians or degrees) and desired precision.
- Compute to get principal value \(\arctan(x)\) or the full-quadrant angle via \(\operatorname{atan2}\).
- (Optional) Expand/simplify with identities, or convert between units.
- Copy the exact form (when available) and the decimal approximation for your work.
Core Formulas (LaTeX)
Definition & range: \[ \theta=\arctan(x)\in\left(-\frac{\pi}{2},\,\frac{\pi}{2}\right),\qquad \tan(\theta)=x. \]
Unit conversions: \[ \theta_{\deg}=\theta_{\text{rad}}\cdot\frac{180}{\pi},\qquad \theta_{\text{rad}}=\theta_{\deg}\cdot\frac{\pi}{180}. \]
Two-argument angle (correct quadrant): \[ \theta=\operatorname{atan2}(y,x)= \begin{cases} \arctan\!\left(\frac{y}{x}\right), & x>0\\[4pt] \arctan\!\left(\frac{y}{x}\right)+\pi, & x<0,\ y\ge0\\[4pt] \arctan\!\left(\frac{y}{x}\right)-\pi, & x<0,\ y<0\\[4pt] \tfrac{\pi}{2}, & x=0,\ y>0\\[4pt] -\tfrac{\pi}{2}, & x=0,\ y<0 \end{cases} \]
Inverse/identity relations: \[ \tan(\arctan x)=x,\qquad \arctan(\tan\theta)=\theta+k\pi \text{ (principal value in }(-\tfrac{\pi}{2},\tfrac{\pi}{2})). \]
Addition formula (branch-aware): \[ \arctan a+\arctan b=\arctan\!\left(\frac{a+b}{1-ab}\right)\ (\text{adjust by }\pm\pi\ \text{when }ab>1\ \text{or quadrants differ}). \]
Derivative: \[ \frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}. \]
Examples (Illustrative)
Example 2 — Slope to angle
Line with slope \(m=\tfrac{3}{4}\): inclination \(\theta=\arctan(\tfrac{3}{4})\approx 0.6435\ \text{rad}\approx36.87^\circ\).
Example 3 — Vector quadrant
For \((x,y)=(-3,-3)\): \(\theta=\operatorname{atan2}(-3,-3)=-\tfrac{3\pi}{4}=-135^\circ\) (third quadrant reference, reported in \((-\pi,\pi]\)).
Example 4 — Combining angles
\(\arctan(2)+\arctan(1)=\arctan\!\big(\tfrac{3}{-1}\big)=-\arctan(3)\). Adjust by \(\pi\) for a positive acute result: \( \pi-\arctan(3)\).
FAQs
What is the range of \(\arctan(x)\)?
It returns principal values in \((-\tfrac{\pi}{2},\,\tfrac{\pi}{2})\), excluding the endpoints.
When should I use \(\operatorname{atan2}(y,x)\) instead of \(\arctan(y/x)\)?
Use \(\operatorname{atan2}\) to get the correct quadrant automatically, especially when \(x\le0\) or \(x=0\).
How do I switch between degrees and radians?
Multiply radians by \(\tfrac{180}{\pi}\) to get degrees and degrees by \(\tfrac{\pi}{180}\) to get radians.
Why doesn’t \(\arctan(\tan\theta)\) always equal \(\theta\)?
Because \(\arctan\) returns principal values; add/subtract multiples of \(\pi\) to map to your desired coterminal angle.
Can the calculator return exact values?
Yes, for recognizable inputs (e.g., \(1,\ \sqrt{3}\)); otherwise it provides high-precision decimal approximations.
What if my input is a fraction or negative?
Fractions are handled exactly; signs are preserved. The angle is negative when the ratio is negative.
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