Inverse Cosine (cos-1) Calculator

Calculate the inverse cosine (arccos) of a value. Input any number from -1 to 1 to find the angle in degrees and radians.

Enter Value

Domain: -1 to 1 inclusive

Result

arccos(0.5)
60°
1.047198 radians
Input value 0.5
Angle (degrees) 60
Angle (radians) 1.047198
sin(theta) 0.866025
tan(theta) 1.732051
Quadrant I
Domain [-1, 1]
Range [0°, 180°] or [0, pi]

Step-by-Step Solution

arccos(0.5) = 60 deg = 1.047198 rad

Understanding Inverse Cosine (Arccos)

The inverse cosine function, denoted as cos-1(x) or arccos(x), answers the question: "What angle has a cosine equal to x?" For example, arccos(0.5) = 60 degrees, because cos(60 degrees) = 0.5.

Since the cosine function is not one-to-one over its entire domain, we must restrict it to make it invertible. The standard restriction is to the interval [0, pi] (0 to 180 degrees), which is the range of the inverse cosine function.

Domain, Range, and Key Properties

Domain

The input value x must be between -1 and 1 inclusive.

Domain: [-1, 1]

Range

The output angle is always between 0 and 180 degrees (0 to pi radians).

Range: [0, pi] or [0 deg, 180 deg]

arccos(-x)

The arccos of a negative value relates to the arccos of the positive value.

arccos(-x) = pi - arccos(x)

Complementary Identity

Arccos and arcsin are complementary (sum to pi/2).

arccos(x) + arcsin(x) = pi/2

Derivative

The derivative of arccos for calculus applications.

d/dx arccos(x) = -1/sqrt(1-x^2)

Inverse Relationship

Arccos undoes cosine within the principal range.

cos(arccos(x)) = x for x in [-1,1]

Common Inverse Cosine Values

These values appear frequently in mathematics and should be memorized:

  • arccos(1) = 0° (0 radians)
  • arccos(√3/2) = 30° (pi/6 radians)
  • arccos(√2/2) = 45° (pi/4 radians)
  • arccos(1/2) = 60° (pi/3 radians)
  • arccos(0) = 90° (pi/2 radians)
  • arccos(-1/2) = 120° (2pi/3 radians)
  • arccos(-√2/2) = 135° (3pi/4 radians)
  • arccos(-√3/2) = 150° (5pi/6 radians)
  • arccos(-1) = 180° (pi radians)

Why is the Range [0, 180 degrees]?

The cosine function is not one-to-one over all angles (it repeats every 360 degrees). To create a proper inverse function, we restrict cosine to the interval [0, pi] where it is strictly decreasing, going from cos(0) = 1 down to cos(pi) = -1. This ensures each output maps to exactly one input.

Applications of Inverse Cosine

  • Triangle solving: Finding angles when sides are known, using the law of cosines: theta = arccos((a^2 + b^2 - c^2) / (2ab)).
  • Dot product angle: The angle between two vectors is theta = arccos(u . v / (|u| |v|)).
  • Physics: Finding the angle of reflection, refraction, or projection.
  • Computer graphics: Calculating lighting angles, surface normals, and rotations.
  • Navigation: Great circle distances and bearing calculations use arccos.

Tips for Using Arccos

  • The input must be between -1 and 1; values outside this range are undefined in real numbers.
  • The result is always in Quadrant I (0-90 degrees) or Quadrant II (90-180 degrees).
  • For positive inputs, the angle is in Quadrant I (0 to 90 degrees).
  • For negative inputs, the angle is in Quadrant II (90 to 180 degrees).
  • arccos(0) = 90 degrees, marking the boundary between Quadrants I and II.