Arccos Calculator (Inverse Cosine)
Calculate the inverse cosine (arccos) of a value and get results in radians and degrees
Calculate Inverse Cosine (arccos)
Domain: -1 ≤ x ≤ 1
Range: 0 to π (or 0° to 180°)
Arccos Results
Function notation: arccos(0) = cos⁻¹(0)
Calculation: arccos(0) = 1.570796 radians = 90.00°
Special angle: This is a common trigonometric value = π/2 radians
Input Domain
The arccos function is only defined for values between -1 and 1 (inclusive).
- -1 ≤ x ≤ 1
- Example: -1, -0.5, 0, 0.5, 1
- 0 ≤ arccos(x) ≤ π radians
- 0° ≤ arccos(x) ≤ 180°
Common Arccos Values
| x | arccos(x) (radians) | arccos(x) (degrees) | Exact Value |
|---|---|---|---|
| -1 | π ≈ 3.14159 | 180° | π |
| -√3/2 ≈ -0.866 | 5π/6 ≈ 2.618 | 150° | 5π/6 |
| -√2/2 ≈ -0.707 | 3π/4 ≈ 2.356 | 135° | 3π/4 |
| -1/2 = -0.5 | 2π/3 ≈ 2.094 | 120° | 2π/3 |
| 0 | π/2 ≈ 1.571 | 90° | π/2 |
| 1/2 = 0.5 | π/3 ≈ 1.047 | 60° | π/3 |
| √2/2 ≈ 0.707 | π/4 ≈ 0.785 | 45° | π/4 |
| √3/2 ≈ 0.866 | π/6 ≈ 0.524 | 30° | π/6 |
| 1 | 0 | 0° | 0 |
Quick Reference
Function
y = arccos(x) or y = cos⁻¹(x)
Domain
-1 ≤ x ≤ 1
Range
0 ≤ y ≤ π (0° to 180°)
Relationship
If y = arccos(x), then cos(y) = x
Key Properties
arccos is the inverse function of cosine
Principal value range: [0, π]
arccos(-x) = π - arccos(x)
arccos(x) is a decreasing function
Graph is reflection of cos(x) over y = x
Applications
Triangle angle calculations (Law of Cosines)
Physics: angle between vectors
Engineering: structural analysis
Navigation and surveying
Computer graphics and 3D modeling
Understanding Arccos (Inverse Cosine)
What is Arccos?
Arccos (or inverse cosine) is the inverse function of the cosine function. It answers the question: "What angle has a cosine value of x?" The notation arccos(x) is equivalent to cos⁻¹(x).
Domain and Range
- •Domain: [-1, 1] - cosine values are always between -1 and 1
- •Range: [0, π] radians or [0°, 180°] - principal value range
- •The function is strictly decreasing within its domain
Mathematical Properties
cos(arccos(x)) = x for x ∈ [-1, 1]
arccos(cos(θ)) = θ for θ ∈ [0, π]
arccos(-x) = π - arccos(x)
Why Principal Values?
Since cosine is periodic (cos(θ) = cos(θ + 2πn)), multiple angles can have the same cosine value. The principal value convention restricts the output to [0, π] to ensure the function is well-defined and one-to-one.
Common Applications
Triangle Solutions
Using the Law of Cosines: if you know all three sides of a triangle, arccos helps find the angles.
Vector Mathematics
The angle between two vectors can be found using arccos of their dot product divided by the product of their magnitudes.
Step-by-Step Example
Problem: Find arccos(0.5)
Step 1: Verify input is in domain [-1, 1] ✓
Step 2: Apply arccos function
Step 3: arccos(0.5) = π/3 radians = 60°
Verification: cos(60°) = cos(π/3) = 0.5 ✓