Inverse Tangent Calculator

Inverse tangent is a trigonometric function that returns the angle whose tangent equals a given number. The inverse tangent is written as tan⁻¹(x), arctan(x), or atan(x). All 3 notations refer to the same inverse trigonometric function. In European and German textbooks this same function appears as arcus tangens or arkus tangens. On the unit circle, arctan returns the angle that corresponds to any tangent value along the circle.

In a right triangle, the tangent of angle θ equals the opposite side divided by the adjacent side. The inverse tangent reverses that operation: given the ratio opposite / adjacent, arctan returns the angle θ.

The domain of the inverse tangent function is all real numbers (−∞, +∞), and the range (principal value) is (−π/2, π/2) in radians or (−90°, 90°) in degrees. The inverse tangent is an odd function, meaning arctan(−x) = −arctan(x).

To calculate the inverse tangent, use the formula θ = arctan(x), where x is the tangent value and θ is the resulting angle.

3 steps summarize the tangent calculation process for the inverse of tangent:

1. Identify the input value (x). The input is any real number—a decimal, a fraction, or the ratio opposite / adjacent from a right triangle.
2. Apply the arctangent function. Use a scientific calculator, a programming language’s atan() function, or Wolfram Alpha to compute tan⁻¹(x).
3. Convert the result. Most math libraries return the answer in radians. Multiply the radian result by 180/π to get degrees, or multiply a degree value by π/180 to convert back to radians. Either way the inverse tan formula remains θ = arctan(x) throughout.

Press the “2nd” or “Shift” key on a scientific calculator (such as Casio or TI–84), then press the “tan” button to access tan⁻¹.

4 steps cover the process on most scientific calculators:

1. Turn on the calculator and set the angle mode to degrees (DEG) or radians (RAD).
2. Press “2nd” or “Shift.” This key activates the secondary functions printed above each button. This shift tan key combination is universal across Casio, TI–84, and Sharp models. Always set your angle mode to DEG or RAD before pressing it, since the same input returns different numbers depending on the mode.
3. Press the “tan” key. The display shows “tan⁻¹(” or “atan(”.
4. Type the value and press “=”. The calculator returns the angle.

Use the arctangent Taylor series expansion to approximate arctan(x) by hand: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … This series converges when |x| ≤ 1.

3 manual methods exist for computing the inverse tangent without a calculator:

1. Taylor series (Maclaurin series). The arctangent series approximation tool uses x − x³/3 + x⁵/5 − x⁷/7. More terms give higher precision. This method works best when |x| ≤ 1.
2. Known angle values. Memorize common results: arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°, arctan(1/√3) = 30°. Use these as reference points.
3. CORDIC algorithm. This hardware–level method computes arctangent through repeated angle rotations using only addition, subtraction, and bit–shifting—common in embedded systems and real–time processing.

The arctan(x) graph is an S–shaped curve that passes through the origin and approaches −π/2 (−90°) as x → −∞ and π/2 (90°) as x → +∞.

3 key features define the arctangent graphical representation:

• Horizontal asymptotes at y = −π/2 and y = +π/2. The function never reaches ±90°.
• Odd symmetry. The graph reflects through the origin because arctan(−x) = −arctan(x).
• Monotonically increasing. The curve always rises from left to right. The arctangent derivative equals 1 / (1 + x²), which stays positive for all x.

The notation for the inverse of tangent takes 3 standard forms: tan⁻¹(x), arctan(x), and atan(x). All three notations describe the exact same mathematical function—the function that accepts a real number and returns the angle whose tangent equals that number.

Each notation appears in different contexts:

• tan⁻¹(x) is the most common notation on scientific calculators, including Casio, TI–84, and Sharp models. The superscript −1 means “inverse function,” not “1 ÷ tan(x).” This distinction frequently confuses students—tan⁻¹(x) returns an angle, whereas 1/tan(x) equals cot(x).
• arctan(x) is the standard notation in formal mathematics, especially in calculus and analysis. The prefix “arc” refers to the arc length on a unit circle that corresponds to the angle. Wolfram Alpha, many university textbooks, and Wikipedia use this form.
• atan(x) is the programming and software notation. JavaScript uses Math.atan(x), Python uses math.atan(x), MATLAB uses atan(x), and Excel/Google Sheets use =ATAN(value). The two–argument variant atan2(y, x) extends the range to (−π, π] for correct quadrant detection in navigation and robotics.

A common pitfall: the notation tan⁻¹(x) does not mean the reciprocal of tangent. The reciprocal of tan(x) is cot(x) = cos(x) / sin(x), which is a completely different function. When you see tan⁻¹ on a calculator or in a textbook, it always refers to the inverse tangent (arctangent) operation.

Some students informally call it anti tangent since it reverses what tangent does, though this is not standard mathematical terminology.

The tan⁻¹ of negative numbers returns a negative angle. Because arctan is an odd function, arctan(−x) = −arctan(x). A negative input always maps to an angle between −90° and 0° (−π/2 and 0 radians).

2 examples of tan⁻¹ of negative numbers:

• arctan(−1) = −45° or −π/4 radians
• arctan(−√3) = −60° or −π/3 radians

What is the tan⁻¹ of −1?

The tan⁻¹ of −1 is −45° (−π/4 radians or −0.7854 radians). This result means an angle of −45° has a tangent value of −1. On the unit circle, this angle corresponds to a point in the fourth quadrant where the opposite side and adjacent side have equal absolute lengths but opposite signs.