Inverse Tangent Calculator

Inverse tangent is the function that returns the angle whose tangent equals a given number. Inverse tangent is written as tan⁻¹(x), arctan(x) or atan(x), and it reverses the tangent function: tan(45°) = 1, so tan⁻¹(1) = 45°.

Inverse tangent (arctangent) is one of the 6 inverse trigonometric functions in trigonometry: arcsin, arccos, arctan, arccsc, arcsec and arccot. Arctangent accepts every real number and returns a principal value between −90° and 90° (−π/2 and π/2 radians).

In a right triangle, the tangent of an angle θ equals the opposite side divided by the adjacent side. The inverse tangent takes that opposite/adjacent ratio and returns the angle θ. In algebra, tan⁻¹ solves equations such as tan θ = 2: θ = tan⁻¹(2) = 63.43°.

• Input: any real number, such as a ratio, a slope or a decimal.
• Output: an angle in degrees or radians.
• Range: −90° < θ < 90°, or −π/2 < θ < π/2.

To calculate inverse tangent, divide the opposite side by the adjacent side and apply tan⁻¹ to the result. The inverse tangent calculation has 3 steps:

1. Divide the opposite side by the adjacent side: x = opposite / adjacent.
2. Apply the inverse tangent: θ = tan⁻¹(x), in radians.
3. Convert the angle to degrees, if needed: degrees = radians × 180/π.

Opposite = 3 and adjacent = 4 give x = 0.75. tan⁻¹(0.75) = 0.6435 rad = 36.87°. The hypotenuse follows from the Pythagorean theorem: √(3² + 4²) = 5.

Two more inverse tangent examples: tan⁻¹(0) = 0° and tan⁻¹(1/3) = 18.43° (0.3218 rad).

To use this inverse tangent calculator, choose an input method, type the value and read the angle. Using the tan-1 calculator takes 5 steps:

1. Choose the input: Decimal / x, Opp / Adj (opposite / adjacent) or Point (x, y).
2. Enter the value: a decimal, a fraction such as 3/4 or an expression such as 1/√3.
3. Pick 1 of 9 result units, including degrees, radians, gradians and minutes or seconds of arc.
4. Set the rounding: integer, tenths, hundredths, thousandths or up to 15 decimal places for maximum accuracy.
5. Read the angle, the exact value, the step-by-step solution and the diagram. Switch on Show labels to name the triangle sides.

This arc tan calculator updates the result as you type. The Point (x, y) input uses atan2(y, x), so the arctan calculator returns the full angle in all 4 quadrants, from −180° to 180°.

The inverse tangent formula is θ = tan⁻¹(opposite / adjacent).

The formula comes from the tangent ratio of a right triangle, tan θ = opposite / adjacent. Applying tan⁻¹ to both sides isolates the angle θ. There are 7 related inverse tangent formulas:

• Coordinates: θ = atan2(y, x) for a point (x, y).
• Slope: θ = tan⁻¹(m), where m = rise / run.
• Complement: tan⁻¹(x) + tan⁻¹(1/x) = 90° for x > 0.
• Sum: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a + b) / (1 − ab)) for ab < 1.
• Derivative (calculus, differentiation): d/dx tan⁻¹(x) = 1 / (1 + x²).
• Integral (integration): ∫ tan⁻¹(x) dx = x·tan⁻¹(x) − ½·ln(1 + x²) + C.
• Complex numbers and hyperbolic functions: tan⁻¹(z) = (i/2)·ln((i + z) / (i − z)) = −i·tanh⁻¹(iz).

Inverse tangent in degrees gives the angle on a scale from −90° to 90°. To convert a radian result to degrees, multiply by 180/π (about 57.2958). This radians to degrees conversion reads θ° = θ rad × 180/π.

A degree splits into degrees, minutes and seconds (DMS). One degree has 60 minutes, and one minute has 60 seconds: tan⁻¹(0.75) = 36.8699° = 36° 52′ 11.6″. For seconds to degrees, divide the seconds by 3,600.

Inverse tangent in radians gives the angle as an arc length on the unit circle, between −π/2 and π/2 (−1.5708 to 1.5708). One radian is the angle whose arc on a circle equals the radius, and a full turn is 2π radians (360°).

For degrees to radians conversion, multiply the degrees by π/180. There are 4 inverse tangent values in radians worth knowing:

• tan⁻¹(1) = π/4 = 0.7854 rad
• tan⁻¹(√3) = π/3 = 1.0472 rad
• tan⁻¹(0.75) = 0.6435 rad
• tan⁻¹(1/3) = 0.3218 rad

MATLAB, Excel, Python and most programming languages return atan results in radians. MATLAB uses atand(x) for degrees, and Excel wraps the result in DEGREES().

To use inverse tangent on a scientific calculator, press the SHIFT or 2nd key, press tan, enter the number and press equals. There are 4 steps:

1. Set the angle mode: DEG for degrees or RAD for radians.
2. Press SHIFT (Casio), 2nd (Texas Instruments) or INV to reach tan⁻¹.
3. Press tan to open tan⁻¹( and type the number.
4. Press = to show the angle.

On a Casio fx-82 or fx-991, the key sequence SHIFT, tan, 1, = returns 45 in degree mode and 0.7853981634 in radian mode. A wrong angle mode is the most common error: tan⁻¹(1) shows 0.785 in RAD mode instead of 45.

Windows Calculator and iPhone Calculator use the reverse order: type the number first, then press tan⁻¹.

To find inverse tangent on Android Calculator (Google Calculator), open the scientific keys, tap INV and tap tan⁻¹. The process has 4 steps:

1. Open Calculator and tap the expand arrow above the keypad, or turn the phone sideways.
2. Tap DEG or RAD to choose degrees or radians.
3. Tap INV: sin, cos and tan change to sin⁻¹, cos⁻¹ and tan⁻¹.
4. Tap tan⁻¹, type the number and tap =. tan⁻¹(1) shows 45.

Samsung Calculator follows the same order: open the scientific keys, tap Inv, then tap tan⁻¹.

To find inverse tangent on iPhone Calculator, switch to scientific mode, type the number, tap 2nd and tap tan⁻¹. The process has 4 steps:

1. Open Calculator. On iOS 18, tap the calculator icon and choose Scientific; on iOS 17 or earlier, turn the iPhone sideways.
2. Check the angle mode: Rad on the display means radians, and the Rad/Deg key switches modes.
3. Type the number, for example 1.
4. Tap 2nd, then tan⁻¹. The display shows 45.

iPhone Calculator applies tan⁻¹ to the number on the screen, so the number comes first.

To find inverse tangent on Windows Calculator, switch to Scientific mode, type the number, open Trigonometry, select 2nd and select tan⁻¹. The process has 4 steps:

1. Open Calculator, select the menu (☰) and choose Scientific.
2. Select the angle mode button until it shows DEG, RAD or GRAD.
3. Type the number, for example 1.
4. Open Trigonometry, select 2nd, then select tan⁻¹. The display shows 45.

The keyboard shortcut for tan⁻¹ in Windows Calculator is Shift + T.

To calculate inverse tangent without a calculator, use 1 of 3 methods: special triangles, a Taylor series or the small-angle rule.

1. Special right triangles give exact values. A 45-45-90 triangle gives tan⁻¹(1) = 45°, and a 30-60-90 triangle gives tan⁻¹(1/√3) = 30° and tan⁻¹(√3) = 60°.
2. The Taylor series tan⁻¹(x) = x − x³/3 + x⁵/5 − x⁷/7 + … works for |x| ≤ 1. For |x| > 1, use tan⁻¹(x) = π/2 − tan⁻¹(1/x) when x > 0. Four terms give tan⁻¹(0.5) ≈ 0.5 − 0.041667 + 0.00625 − 0.001116 = 0.463467 rad, close to the true 0.463648 rad (26.565°).
3. The small-angle rule tan⁻¹(x) ≈ x radians stays within 0.4% for |x| ≤ 0.1.

Calculators and computer chips evaluate arctangent with the CORDIC algorithm (COordinate Rotation DIgital Computer). CORDIC rotates a vector through fixed small angles with shifts and additions only.

The inverse of tangent has 4 common notations: tan⁻¹(x), arctan(x), atan(x) and arctg(x).

The −1 in tan⁻¹(x) marks an inverse function, not an exponent. tan⁻¹(x) differs from (tan x)⁻¹ = 1/tan(x), which equals cot(x). The arc prefix comes from the unit circle: arctan(x) is the arc length, in radians, whose tangent is x.

The arctan(x) graph is an increasing S-shaped curve through the origin that levels off at two horizontal asymptotes, y = π/2 and y = −π/2. The inverse tangent graph has 5 key features:

• Domain: all real numbers, (−∞, ∞).
• Range: (−π/2, π/2), or (−90°, 90°).
• Symmetry: an odd function, symmetric about the origin.
• Slope: 1 at x = 0, from the derivative 1 / (1 + x²).
• Inflection point: (0, 0).

The arctan graph is the reflection of the tangent graph on (−π/2, π/2) across the line y = x. The domain and range of the two graphs swap places.

The inverse tangent table lists tan⁻¹(x) for common values of x in degrees and radians. Exact values come from the special angles 15°, 30°, 45°, 60° and 75°.

The tangent table runs the other way, from angle to ratio: a tangent calculation gives tan(30°) = 0.5774, tan(45°) = 1 and tan(60°) = 1.7321. Large inputs approach 90°: tan⁻¹(45) = 88.73° and tan⁻¹(100) = 89.43°.

Select a row to load the value into the calculator. Use the rounding buttons to switch between integer, tenths, hundredths, thousandths and maximum accuracy.

To find tan⁻¹ of negative numbers, find tan⁻¹ of the positive value and put a minus sign in front: tan⁻¹(−x) = −tan⁻¹(x). Arctangent is an odd function, so a negative input always gives an angle between −90° and 0°.

• tan⁻¹(−0.5) = −26.57°
• tan⁻¹(−√3) = −60°
• tan⁻¹(−2) = −63.43°

A negative principal value points into quadrant IV. Add 180° for a quadrant II angle, when the point lies left of the y-axis: atan2(1, −1) = 135°.

What is the tan−1 of −1?

The tan⁻¹ of −1 is −45°, or −π/4 radians (−0.7854 rad). tan(−45°) = −1, and −45° lies inside the principal range of −90° to 90°. On a 0° to 360° scale, the same direction is 315°.