Inverse Cosine Calculator

Inverse cosine is the function that returns the angle whose cosine equals a given number. Inverse cosine is written as cos⁻¹(x), arccos(x) or acos(x), and it reverses the cosine function: cos(60°) = 0.5, so cos⁻¹(0.5) = 60°.

Inverse cosine (arccosine) is one of the 6 inverse trigonometric functions. Arccosine accepts numbers from −1 to 1 and returns a principal value from 0° to 180° (0 to π radians). The range starts at 0°, not at −90° like arcsine, so arccos never returns a negative angle.

In a right triangle, the cosine of an angle θ equals the adjacent side divided by the hypotenuse. The inverse cosine takes that adjacent/hypotenuse ratio and returns θ. For a ramp 4 m long that covers 3.2 m of ground, θ = cos⁻¹(3.2/4) = cos⁻¹(0.8) = 36.87°.

• Input: a number from −1 to 1, such as a cosine value or a side ratio.
• Output: an angle from 0° to 180°.
• Special values: cos⁻¹(1) = 0°, cos⁻¹(0) = 90°, cos⁻¹(−1) = 180°.

The −1 in cos⁻¹(x) marks the inverse function, not an exponent. The reciprocal 1/cos(x) is the secant sec(x).

To calculate inverse cosine, divide the adjacent side by the hypotenuse, then apply cos⁻¹ to the result. Follow these 4 steps:

1. Find the cosine value x. For a right triangle, x = adjacent ÷ hypotenuse. Example: 12 ÷ 13 = 0.9231.
2. Check that x lies from −1 to 1. A value outside that interval has no real arccosine.
3. Apply the inverse cosine: θ = cos⁻¹(0.9231) = 22.62°.
4. Convert the unit if needed: 22.62° × π/180 = 0.3948 rad.

Check the answer with the cosine: cos(22.62°) = 0.9231, which matches x.

A shortcut through arcsine: cos⁻¹(x) = 90° − sin⁻¹(x). For x = 0.6, sin⁻¹(0.6) = 36.87°, so cos⁻¹(0.6) = 90° − 36.87° = 53.13°.

Exact angles: cos⁻¹(√3/2) = 30° = π/6, cos⁻¹(√2/2) = 45° = π/4, cos⁻¹(1/2) = 60° = π/3, cos⁻¹(0) = 90° = π/2 and cos⁻¹(−1/2) = 120° = 2π/3.

Enter a value from −1 to 1, or the adjacent side and hypotenuse, and the calculator returns cos⁻¹(x) as you type. The steps:

1. Pick an input method: Decimal / x for a cosine value, or Adj / Hyp for 2 sides of a right triangle.
2. Type the value. The field accepts decimals (0.8), fractions (4/5), roots (√3/2), π and expressions.
3. Choose the result unit, from degrees and radians to gradians, turns, arcminutes, arcseconds, milliradians, microradians and multiples of π.
4. Choose the rounding, from whole numbers to 15 decimal places.
5. Read the angle in all 9 units, the exact value, the triangle, the unit circle, the graph and the worked steps.

The calculator rejects values outside −1 to 1 and states the reason. In Adj / Hyp mode, the adjacent side must not exceed the hypotenuse. Results can be copied, shared as a link, printed or kept in the history list.

3 sides mode uses the law of cosines on any triangle, not only right triangles. Enter sides a, b and c; the calculator returns the angle C opposite c, plus angles A and B and the area. Sides 5, 6 and 7 give C = arccos((25 + 36 − 49)/60) = arccos(0.2) = 78.46°.

The inverse cosine formula is θ = cos⁻¹(adjacent / hypotenuse). θ is the angle, the adjacent side touches θ, and the hypotenuse is the longest side.

Related forms of the formula:

• Identity: cos(cos⁻¹ x) = x for −1 ≤ x ≤ 1, and cos⁻¹(cos θ) = θ for 0° ≤ θ ≤ 180°.
• Complement rule: cos⁻¹ x = 90° − sin⁻¹ x.
• Law of cosines: C = cos⁻¹((a² + b² − c²) / 2ab) gives any angle of any triangle from its 3 sides.
• Dot product: the angle between vectors u and v is θ = cos⁻¹(u·v / (|u||v|)).
• Derivative: d/dx cos⁻¹ x = −1/√(1 − x²).
• Integral: ∫ cos⁻¹ x dx = x·cos⁻¹ x − √(1 − x²) + C.

Example: a 5 m ladder stands 3 m from a wall. cos θ = 3/5 = 0.6, so the ladder meets the ground at θ = cos⁻¹(0.6) = 53.13°.

Inverse cosine in degrees returns an angle from 0° to 180°. A calculator in DEG mode gives cos⁻¹(0.5) = 60°, cos⁻¹(0.8) = 36.87° and cos⁻¹(−0.8) = 143.13°.

Positive inputs give acute angles (0° to 90°); negative inputs give obtuse angles (90° to 180°). The split point is cos⁻¹(0) = 90°.

To convert a radian result to degrees, multiply by 180/π: cos⁻¹(0.3) = 1.2661 rad × 57.2958 = 72.54°. In degrees, minutes and seconds, 72.5424° = 72° 32′ 32.6″.

Common values in degrees:

• cos⁻¹(1) = 0°
• cos⁻¹(√3/2) = 30°
• cos⁻¹(√2/2) = 45°
• cos⁻¹(1/2) = 60°
• cos⁻¹(0) = 90°
• cos⁻¹(−1/2) = 120°
• cos⁻¹(−1) = 180°

Inverse cosine in radians returns an angle from 0 to π (0 to 3.1416). cos⁻¹(0.5) = π/3 = 1.0472 rad, cos⁻¹(0) = π/2 = 1.5708 rad and cos⁻¹(−1) = π = 3.1416 rad.

On the unit circle, cos⁻¹(x) is the arc length from (1, 0) to the point of the upper half circle whose x-coordinate equals x. The cosine is the horizontal coordinate, so the point slides left as x falls from 1 to −1.

To convert degrees to radians, multiply by π/180: 120° × π/180 = 2π/3. Excel, Python, JavaScript, C and Java return arccosine in radians by default.

• cos⁻¹(√3/2) = π/6 = 0.5236 rad
• cos⁻¹(√2/2) = π/4 = 0.7854 rad
• cos⁻¹(1/2) = π/3 = 1.0472 rad
• cos⁻¹(−1/2) = 2π/3 = 2.0944 rad
• cos⁻¹(−√2/2) = 3π/4 = 2.3562 rad

The domain of inverse cosine is −1 ≤ x ≤ 1, and the range is 0° ≤ θ ≤ 180° (0 ≤ θ ≤ π).

The cosine of a real angle lies from −1 to 1, so cos⁻¹(1.2) and cos⁻¹(−3) have no real value.

Cosine repeats every 360°, and cos(60°) = cos(300°) = cos(−60°) = 0.5. Inverse cosine restricts cosine to 0° to 180°, where cosine falls from 1 to −1 and takes each value once. That interval gives the principal value.

• Domain: [−1, 1]
• Range: [0, π]
• Monotonic: decreasing over the whole domain
• Symmetry: cos⁻¹(−x) = π − cos⁻¹(x)
• End points: cos⁻¹(1) = 0° and cos⁻¹(−1) = 180°

To find every angle with cos θ = x, use θ = ±cos⁻¹(x) + 360°·k for any integer k. For cos θ = 0.5 in [0°, 360°), the answers are 60° and 300°.

On a scientific calculator, press SHIFT (or 2nd), then cos, type the value and press =. On a Casio fx-991 or fx-82, the keys for cos⁻¹(0.5) are SHIFT, cos, 0.5, ), = and the screen shows 60.

1. Set the angle mode: D on the screen means degrees, R means radians.
2. Press SHIFT (Casio, Sharp) or 2nd (Texas Instruments) to reach cos⁻¹ above the cos key.
3. Type the value, close the bracket and press =.

On a TI-84, press 2nd, COS, 0.5, ENTER. Press MODE to switch between DEGREE and RADIAN.

Math ERROR (Casio) or ERR:DOMAIN (TI) means the value lies outside −1 to 1. A result of 1.0472 in place of 60 means the calculator is in radian mode.

On the Android (Google) Calculator, open the scientific keys, tap INV, tap cos⁻¹, type the value and tap =. cos⁻¹(0.5) returns 60 in DEG mode.

1. Tap the expand arrow, or turn the phone to landscape, to show the scientific keys.
2. Check the mode label: DEG for degrees, RAD for radians.
3. Tap INV. The cos key changes to cos⁻¹.
4. Tap cos⁻¹, type 0.5 and tap =.

In RAD mode, the same keys return 1.0471975512, which is π/3.

On the iPhone Calculator, switch to Scientific, type the value, tap 2nd, then tap cos⁻¹. Type 0.5, tap 2nd and tap cos⁻¹: the display shows 60.

1. Turn the iPhone sideways, or tap the calculator icon and choose Scientific (iOS 18 and later).
2. Check the mode: Rad on the display means radians, no label means degrees.
3. Type 0.5.
4. Tap 2nd. The cos key changes to cos⁻¹.
5. Tap cos⁻¹. The result appears at once.

The iPhone applies cos⁻¹ to the number on the display, so type the value first.

In Windows Calculator, choose Scientific, type the value, open Trigonometry, select 2nd, then select cos⁻¹. For 0.5 in DEG mode, the result is 60.

1. Open the menu (☰) and choose Scientific.
2. Check the angle button: DEG, RAD or GRAD.
3. Type 0.5.
4. Open Trigonometry and select 2nd.
5. Select cos⁻¹. The result is 60.

Keyboard shortcut: type 0.5, then press Shift + O for cos⁻¹. F3, F4 and F5 switch to degrees, radians and gradians.

Software calls inverse cosine ACOS or acos, and the result comes out in radians. Convert the result to get degrees.

• Excel and Google Sheets: =ACOS(0.5) returns 1.0472; =DEGREES(ACOS(0.5)) returns 60.
• Python: math.acos(0.5) returns 1.0472; math.degrees(math.acos(0.5)) returns 60.00000000000001. NumPy uses np.arccos.
• JavaScript: Math.acos(0.5) * 180 / Math.PI returns 60.00000000000001.
• MATLAB and Octave: acos(0.5) for radians, acosd(0.5) for degrees.
• C, C++ and Java: acos(0.5) and Math.acos(0.5) return radians.
• Wolfram Alpha: ArcCos[1/2] returns π/3.

Values outside −1 to 1 produce errors: Excel returns #NUM!, Python raises ValueError: math domain error, and JavaScript returns NaN. Floating-point inputs a hair above 1, such as 1.0000000000000002 from a dot product, trigger the same error, so clamp the value to [−1, 1] first.

Without a calculator, use exact values for special angles, or subtract the arcsine series from 90°.

1. Special values: cos 30° = √3/2, cos 45° = √2/2 and cos 60° = 1/2, so cos⁻¹(√3/2) = 30°, cos⁻¹(√2/2) = 45° and cos⁻¹(1/2) = 60°.
2. Series: cos⁻¹ x = π/2 − (x + x³/6 + 3x⁵/40 + 5x⁷/112 + …) for |x| ≤ 1.
3. Negative inputs: compute cos⁻¹ of the positive value, then subtract from 180°: cos⁻¹(−0.5) = 180° − 60° = 120°.

Example: cos⁻¹(0.3) ≈ 1.570796 − (0.3 + 0.0045 + 0.000182 + 0.0000098) = 1.266104 rad = 72.542°. The true value is 1.266104 rad.

The series converges slowly near x = ±1. For x above 0.7, use cos⁻¹ x = sin⁻¹(√(1 − x²)) to work with a smaller number.

The graph of y = cos⁻¹(x) is a decreasing curve from (−1, π) to (1, 0) that crosses the y-axis at (0, π/2).

• Starts at (−1, 180°), passes (0, 90°) and ends at (1, 0°).
• Slope −1 at x = 0; the slope −1/√(1 − x²) falls without bound near x = ±1.
• Point symmetry about (0, π/2), the inflection point.
• Mirror image of y = cos x on [0, π] across the line y = x.
• The graph of cos⁻¹ x is the graph of sin⁻¹ x flipped upside down and shifted up by π/2.

The curve stays above the x-axis everywhere: every arccosine value is 0 or positive.

The inverse cosine table lists cos⁻¹(x) for common values of x from −1 to 1, in degrees and radians. Click a row to load it into the calculator.

Exact values to know: cos⁻¹(1) = 0, cos⁻¹(√3/2) = π/6, cos⁻¹(√2/2) = π/4, cos⁻¹(1/2) = π/3, cos⁻¹(0) = π/2 and cos⁻¹(−1) = π.

Each negative row equals 180° minus its positive partner: cos⁻¹(−0.6) = 180° − 53.13° = 126.87°.

The inverse cosine of a negative number is an obtuse angle: cos⁻¹(−x) = 180° − cos⁻¹(x). cos⁻¹(−0.5) = 120°, cos⁻¹(−√2/2) = 135° and cos⁻¹(−1) = 180°.

Arccosine is not an odd function, so a minus sign does not carry through: cos⁻¹(−0.5) is 120°, not −60°. The angle lands in quadrant II, from 90° to 180°, where cosine is negative.

What is cos⁻¹(−0.5)?

cos⁻¹(−0.5) = 120° = 2π/3 rad. cos(120°) = −0.5, and 120° lies inside the range 0° to 180°. The angle 240° has the same cosine, −0.5, but sits outside the principal range.

arccos takes adjacent/hypotenuse and returns 0° to 180°; arcsin takes opposite/hypotenuse and returns −90° to 90°; arctan takes opposite/adjacent and accepts any real number.

In a 3-4-5 triangle, the angle next to the side of 4 is cos⁻¹(4/5) = sin⁻¹(3/5) = tan⁻¹(3/4) = 36.87°. All 3 functions agree on acute angles of a right triangle.

The identity sin⁻¹ x + cos⁻¹ x = 90° holds for every x from −1 to 1. The secant inverse links to arccosine: sec⁻¹ x = cos⁻¹(1/x) for |x| ≥ 1.

Inverse cosine finds angles from side lengths and from dot products, which makes it the standard tool for triangles that are not right triangles and for angles between directions.

• Law of cosines: a triangle with sides 5, 6 and 7 has the angle C = cos⁻¹((5² + 6² − 7²) / (2·5·6)) = cos⁻¹(0.2) = 78.46° opposite the side of 7.
• Vectors: the angle between (3, 4) and (4, 0) is cos⁻¹(12 / (5·4)) = cos⁻¹(0.6) = 53.13°. Games and 3D graphics use this for field-of-view and lighting checks.
• AC circuits: a load whose ratio of real to apparent watts (cos φ) is 0.8 has a phase angle φ = cos⁻¹(0.8) = 36.87° between voltage and current.
• Surveying and navigation: the great-circle distance formula uses cos⁻¹ to turn 2 latitude/longitude pairs into a central angle.
• Robotics: joint angles of a 2-link arm come from the law of cosines.