Inverse Tangent Calculator (arctan)
Enter any real number to find its inverse tangent (arctan). The result appears in both degrees and radians. Switch to Right Triangle mode to find an angle from opposite and adjacent side lengths. The Show Your Work panel walks through every step of the calculation.
Formula
Worked example
Find the angle whose tangent is 1.732 (approximately sqrt(3)). arctan(1.732) = 60 degrees = pi/3 radians. Verification: tan(60 deg) = 1.7321, which matches. Using the right-triangle identity: sin(arctan(1.732)) = 1.732 / sqrt(1 + 1.732^2) = 1.732 / 2 = 0.866, which equals sin(60 deg).
What is the inverse tangent (arctan)?
The inverse tangent, written arctan(x) or tan^(-1)(x), answers the question: "which angle has a tangent equal to x?" Because the ordinary tangent function repeats every 180 degrees (pi radians), the inverse needs a restricted range to give a unique answer. By convention, arctan always returns a value in the open interval from -90 to 90 degrees (-pi/2 to pi/2 radians). This principal-value range means every real number maps to exactly one angle. For example, arctan(1) = 45 degrees, arctan(-1) = -45 degrees, and arctan(0) = 0 degrees.
Right-triangle interpretation
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. The inverse tangent reverses that: if you know the two legs of a right triangle, arctan(opposite / adjacent) gives you the angle at that corner. For example, a triangle with opposite leg 3 and adjacent leg 4 has tan(theta) = 3/4 = 0.75, so theta = arctan(0.75) = 36.87 degrees. This is one of the most common uses of arctan in geometry, surveying, navigation, and engineering.
Useful arctan identities
Several identities let you convert arctan results into other trig values without computing a new trig function. If theta = arctan(x), then: sin(theta) = x / sqrt(1 + x^2), and cos(theta) = 1 / sqrt(1 + x^2). These follow directly from a unit-hypotenuse right triangle where the opposite leg is x and the adjacent leg is 1, giving a hypotenuse of sqrt(1 + x^2). The addition formula is arctan(a) + arctan(b) = arctan((a + b) / (1 - ab)) when ab < 1. The complementary identity is arctan(x) + arctan(1/x) = pi/2 for positive x.
Degrees vs. radians: which should you use?
Degrees are the natural choice when communicating angles to people (a 45-degree slope is intuitive). Radians are preferred in mathematics and physics because calculus formulas for sin, cos and tan are simpler when angles are measured in radians, and radian measure makes the arc-length formula s = r*theta dimensionally clean. Most scientific calculators have a DEG/RAD mode switch. When using arctan on a calculator or in code, confirm the mode first: Math.atan() in JavaScript always returns radians, while atan() in SQL varies by implementation. To convert: degrees = radians x (180/pi) and radians = degrees x (pi/180).
Common inverse tangent values
| x (tangent) | Angle (degrees) | Angle (radians) | Exact form |
|---|---|---|---|
| -infinity | -90 | -pi/2 | -pi/2 |
| -sqrt(3) = -1.7321 | -60 | -pi/3 | -pi/3 |
| -1 | -45 | -pi/4 | -pi/4 |
| -1/sqrt(3) = -0.5774 | -30 | -pi/6 | -pi/6 |
| 0 | 0 | 0 | 0 |
| 1/sqrt(3) = 0.5774 | 30 | pi/6 | pi/6 |
| 1 | 45 | pi/4 | pi/4 |
| sqrt(3) = 1.7321 | 60 | pi/3 | pi/3 |
| +infinity | 90 | pi/2 | pi/2 |
Standard arctan reference values you will encounter in trigonometry and geometry.
Frequently asked questions
What is the difference between tan^(-1)(x) and 1/tan(x)?
tan^(-1)(x) is the inverse tangent function (arctan), which returns an angle. 1/tan(x) is the reciprocal of the tangent, also called the cotangent (cot). They are completely different: arctan(1) = 45 degrees, but 1/tan(1 radian) = 0.6421. The superscript -1 on a trig function always means "inverse function", not "raise to the power of -1".
Why does arctan only return values between -90 and 90 degrees?
The tangent function is not one-to-one over its full domain because tan(x) = tan(x + 180 deg) for any x. To define a proper inverse, mathematicians restrict the range to the interval (-90, 90) degrees, called the principal value. Every real number maps to exactly one angle in that interval. If you need an angle outside that range, you need extra information (usually the signs of both x and y coordinates), which is what the atan2 function provides.
How do I compute arctan without a calculator?
For the standard values (0, 30, 45, 60 degrees), you can memorise the table: arctan(0)=0, arctan(1/sqrt(3))=30 deg, arctan(1)=45 deg, arctan(sqrt(3))=60 deg. For other values, the Taylor series arctan(x) = x - x^3/3 + x^5/5 - ... converges for |x| <= 1. For |x| > 1, use the identity arctan(x) = pi/2 - arctan(1/x) first to bring the argument into the convergent range, then apply the series.
How is arctan used in right-triangle problems?
If you know the lengths of the two legs of a right triangle (opposite and adjacent), the angle at the corner between the hypotenuse and the adjacent leg is theta = arctan(opposite / adjacent). For example, a ramp that rises 1.5 m over a horizontal run of 4 m has an angle of arctan(1.5/4) = arctan(0.375) = 20.56 degrees. This application appears in construction, road design, surveying, and navigation.
What happens to arctan as x approaches infinity?
As x approaches positive infinity, arctan(x) approaches 90 degrees (pi/2 radians) but never reaches it. As x approaches negative infinity, arctan(x) approaches -90 degrees (-pi/2 radians). These are horizontal asymptotes of the arctan function. The tangent function grows without bound as its argument nears 90 degrees, so the inverse approaches that boundary but cannot cross it.
What is the derivative of arctan(x)?
d/dx [arctan(x)] = 1 / (1 + x^2). This is a standard result in calculus. The derivative is always positive (arctan is strictly increasing) and is largest at x = 0 (where it equals 1) and decreases toward zero as |x| grows large, matching the flattening of the arctan curve near its asymptotes.