Cos⁻¹ Calculator
Calculate the inverse cosine (arccos) of any value in the domain [-1, 1]
Calculate cos⁻¹(x)
Domain: -1 ≤ x ≤ 1
Range: [0, π] rad or [0°, 180°]
Result
1.570796 rad
cos⁻¹(0) = 1.570796 radians
Exact Value
cos⁻¹(0) = π/2
Radians: π/2
Degrees: 90°
1.570796 rad
Radians
90.000000°
Degrees
Step-by-Step Solution
1.Given: cos⁻¹(0)
2.Domain check: -1 ≤ 0 ≤ 1 ✓
3.Using the inverse cosine function: arccos(0)
4.This is a special angle: cos⁻¹(0) = π/2
5.Result: π/2 radians
Common Values
| x | cos⁻¹(x) (rad) | cos⁻¹(x) (°) | Exact Value |
|---|---|---|---|
| 1 | 0 | 0° | 0 |
| √3/2 ≈ 0.866 | 0.524 | 30° | π/6 |
| √2/2 ≈ 0.707 | 0.785 | 45° | π/4 |
| 1/2 = 0.5 | 1.047 | 60° | π/3 |
| 0 | 1.571 | 90° | π/2 |
| -1/2 = -0.5 | 2.094 | 120° | 2π/3 |
| -√2/2 ≈ -0.707 | 2.356 | 135° | 3π/4 |
| -√3/2 ≈ -0.866 | 2.618 | 150° | 5π/6 |
| -1 | 3.142 | 180° | π |
Function Properties
Domain
[-1, 1]
Range
[0, π] rad or [0°, 180°]
Notation
cos⁻¹(x), arccos(x)
Type
Inverse trigonometric function
Quick Reference
•
cos⁻¹(1) = 0 (minimum value)
•
cos⁻¹(0) = π/2 or 90°
•
cos⁻¹(-1) = π or 180° (maximum)
•
Function is decreasing
•
Not antisymmetric
Understanding the Inverse Cosine Function
What is cos⁻¹(x)?
The inverse cosine function, denoted as cos⁻¹(x) or arccos(x), finds the angle whose cosine equals the given value. It answers the question: "What angle has a cosine of x?"
Important Notes
- •Domain is restricted to [-1, 1] because cosine values are always between -1 and 1
- •Range is [0, π] radians or [0°, 180°] to ensure the function is one-to-one
- •The function is strictly decreasing
Relationship to Cosine
If y = cos⁻¹(x), then cos(y) = x
The inverse cosine "undoes" the cosine function within its restricted domain.
Applications
- •Finding angles in triangles
- •Solving trigonometric equations
- •Physics and engineering calculations
- •Computer graphics and robotics
Common Misconceptions
Warning: cos⁻¹(x) has two different meanings in mathematics:
- 1. Inverse function: arccos(x) - what this calculator computes
- 2. Reciprocal: 1/cos(x) = sec(x) - this is NOT what cos⁻¹ means here