Inverse Cosine Calculator

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Enter a value between -1 and 1 (the cosine of the angle)

Calculate the inverse cosine (arccos) of any value between -1 and 1. Get the angle in degrees and radians with step-by-step results. Essential for geometry and trigonometry.

Arccos Result

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Sources & Methodology

✓ Formulas verified against authoritative sources listed below.

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NIST Digital Library of Mathematical Functions — Chapter 4: Trigonometric Functions

NIST authoritative reference for inverse trigonometric functions including arccos definition and properties

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Wolfram MathWorld — Inverse Cosine

Comprehensive mathematical reference for arccosine properties, identities, and applications

Methodology: arccos(x) = inverse of cosine function. Domain: -1 to 1. Range: 0 to 180 degrees (0 to pi radians). Computed using JavaScript Math.acos(x). Conversion: degrees = radians x (180/pi). Gradians = degrees x (10/9). Verification: cos(arccos(x)) = x.

⏱ Last reviewed: April 2026

How to Calculate the Inverse Cosine

The inverse cosine function (arccos, also written cos⁻¹ or acos) answers the question: given a cosine value, what angle produces it? If cos(60°) = 0.5, then arccos(0.5) = 60°. The arccos function has a domain of -1 to 1 and a principal value range of 0° to 180° (0 to π radians).

The Arccos Function Defined

arccos(x) returns the angle θ such that cos(θ) = x. The principal value range is 0° to 180° (or 0 to π radians). For example: arccos(1) = 0°, arccos(0) = 90°, arccos(-1) = 180°, arccos(0.5) = 60°, arccos(-0.5) = 120°. Values outside -1 to 1 are undefined.

Arccos vs. 1/cos (Secant)

arccos(x) is the inverse cosine function, NOT the reciprocal of cosine. The reciprocal of cosine is secant: sec(θ) = 1/cos(θ). These are fundamentally different: arccos(0.5) = 60° (an angle), while 1/cos(60°) = 1/0.5 = 2 (a ratio). The notation cos⁻¹ is sometimes confused with 1/cos but always means the inverse function in this context.

Applications of Inverse Cosine

Arccos appears in: 1) Triangle solving — finding an angle when two sides are known (law of cosines: cos A = (b² + c² − a²) / (2bc)). 2) Dot product angle formula: angle between vectors = arccos(a·b / |a||b|). 3) Physics — angle of incidence/reflection, pendulum motion, circular motion. 4) Computer graphics — calculating light angles and surface normals.

Unit Conversion: Degrees, Radians, Gradians

Degrees: most common in everyday use. Radians: used in calculus and physics (SI standard). Gradians (grades): used in surveying (400 grad = full circle). Conversions: radians = degrees x π/180. Gradians = degrees x 10/9. For arccos, the result range in radians is 0 to π (about 0 to 3.14159).

arccos(x) = angle where cos(angle) = x | Domain: -1 to 1 | Range: 0 to 180 degrees (0 to pi rad)

Conversion: degrees = radians x (180/pi). Radians = degrees x (pi/180). Verification: cos(arccos(x)) = x always. Common values: arccos(0) = 90 deg, arccos(1) = 0 deg, arccos(-1) = 180 deg, arccos(sqrt(2)/2) = 45 deg.

Inverse Cosine Common Values

x value	arccos(x) degrees	arccos(x) radians	Notes
-1	180°	π (3.14159)	cos(180°) = -1
-0.866	150°	5π/6 (2.61799)	cos(150°) = -sqrt(3)/2
-0.707	135°	3π/4 (2.35619)	cos(135°) = -sqrt(2)/2
-0.500	120°	2π/3 (2.09440)	cos(120°) = -1/2
0	90°	π/2 (1.57080)	cos(90°) = 0
0.500	60°	π/3 (1.04720)	cos(60°) = 1/2
0.707	45°	π/4 (0.78540)	cos(45°) = sqrt(2)/2
0.866	30°	π/6 (0.52360)	cos(30°) = sqrt(3)/2
1	0°	0	cos(0°) = 1

💡 Calculator Tip: When using a physical scientific calculator, the inverse cosine button is usually labeled cos⁻¹ or acos, accessed via the SHIFT or 2nd key + the cos key. Always verify your calculator is in the correct angle mode (DEG or RAD) before computing arccos — the same x value gives very different numeric results in degrees vs. radians.

Frequently Asked Questions

Inverse cosine (arccos) is the function that returns the angle whose cosine equals a given value. If cos(60 degrees) = 0.5, then arccos(0.5) = 60 degrees. Domain: -1 to 1. Range: 0 to 180 degrees (0 to pi radians).

Use the arccos button on a scientific calculator (SHIFT + cos or 2nd + cos). In programming: Math.acos(x) returns radians. Convert to degrees by multiplying by 180/pi. For arccos(0.5): 0.5 radians x 180/pi = 60 degrees.

arccos(0) = 90 degrees = pi/2 radians. This is because cos(90 degrees) = 0.

arccos(1) = 0 degrees = 0 radians. cos(0 degrees) = 1.

arccos(-1) = 180 degrees = pi radians. cos(180 degrees) = -1.

arccos(0.5) = 60 degrees = pi/3 radians. cos(60 degrees) = 0.5.

arccos(x) is the inverse of cosine. arcsec(x) is the inverse of secant (1/cosine). arccos(x) takes values from -1 to 1 and returns 0 to 180 degrees. arcsec(x) takes values outside (-1,1) and returns 0 to 180 degrees excluding 90 degrees.

Domain of arccos: -1 to 1 (any x outside this range is undefined). Range of arccos: 0 to 180 degrees (0 to pi radians). These are the principal values. The function is one-to-one on this restricted range.

Law of cosines: a^2 = b^2 + c^2 - 2bc cos(A). To find angle A: cos(A) = (b^2 + c^2 - a^2) / (2bc). Then A = arccos((b^2 + c^2 - a^2) / (2bc)). This finds angles in any triangle given all three sides.

arccos(x) + arcsin(x) = 90 degrees (pi/2 radians) for all x in -1 to 1. So arccos(x) = 90 degrees - arcsin(x). Also: arccos(x) = pi/2 - arcsin(x) in radians.

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