Understanding the Arccos Function (Inverse Cosine)
The arccos function, written as arccos(x) or cos-1(x), is the inverse of the cosine function. Given a cosine value, it returns the angle whose cosine is that value.
Domain and Range
Domain of arccos: [-1, 1]
Range of arccos: [0, π] radians = [0°, 180°]
If θ = arccos(x), then cos(θ) = x and 0 ≤ θ ≤ π
Range of arccos: [0, π] radians = [0°, 180°]
If θ = arccos(x), then cos(θ) = x and 0 ≤ θ ≤ π
This restricted range is the principal value. On [0, π], cosine is strictly decreasing, making the inverse unique.
Common Arccos Values
| x | Degrees | Radians | Exact |
|---|---|---|---|
| -1 | 180° | 3.14159... | π |
| -√3/2 | 150° | 2.61799... | 5π/6 |
| -√2/2 | 135° | 2.35619... | 3π/4 |
| -1/2 | 120° | 2.09440... | 2π/3 |
| 0 | 90° | 1.57080... | π/2 |
| 1/2 | 60° | 1.04720... | π/3 |
| √2/2 | 45° | 0.78540... | π/4 |
| √3/2 | 30° | 0.52360... | π/6 |
| 1 | 0° | 0 | 0 |
Relationship with Arcsin
arccos(x) + arcsin(x) = π/2 (for all x in [-1, 1])
arccos(x) = π/2 - arcsin(x)
arccos(-x) = π - arccos(x)
arccos(x) = π/2 - arcsin(x)
arccos(-x) = π - arccos(x)
Derivative and Integral
d/dx [arccos(x)] = -1 / sqrt(1 - x²) for |x| < 1
∫ arccos(x) dx = x · arccos(x) - sqrt(1 - x²) + C
∫ arccos(x) dx = x · arccos(x) - sqrt(1 - x²) + C
Applications
- Vector Angles: The angle between two vectors uses arccos of their normalized dot product.
- Navigation: Great circle distances on a sphere use arccos.
- Computer Graphics: Lighting and reflection angle calculations.
- Signal Processing: Phase angle calculations in frequency analysis.
- Robotics: Inverse kinematics for joint angles.