Cos Inverse Calculator
Calculate arccos(x) with step-by-step solutions and exact values
Inverse Cosine Calculator
Domain: -1 ≤ x ≤ 1
Results
Exact Value: arccos(0) = 90° = π/2 rad
Verification: cos(90.0000°) = 0.000000
Complementary angle: 0.0000° = 0.0000 rad
Step-by-Step Solution
1. Given: x = 0
2. Apply inverse cosine function: arccos(x)
3. arccos(0) = 1.570796 radians
4. Convert to degrees: 1.570796 × (180/π) = 90.0000°
5. Exact value: arccos(0) = 90° = π/2 rad
Example Calculations
Special Value: x = 0.5
Given: x = 0.5
Calculation: arccos(0.5)
Result: 60° = π/3 rad
Exact value: arccos(1/2) = 60°
Negative Value: x = -0.5
Given: x = -0.5
Formula: arccos(-x) = π - arccos(x)
Calculation: π - arccos(0.5) = π - π/3
Result: 120° = 2π/3 rad
Common Arccos Values
Arccos Properties
Domain: [-1, 1]
Range: [0, π] rad or [0°, 180°]
Decreasing function
arccos(-x) = π - arccos(x)
cos(arccos(x)) = x
Quick Tips
Always check that input is in [-1, 1]
Result is always between 0° and 180°
For negative inputs, use the symmetry property
Memorize special values for faster calculation
Understanding the Inverse Cosine Function
What is Arccos?
The inverse cosine function, denoted as arccos(x) or cos⁻¹(x), is the inverse of the cosine function. It finds the angle whose cosine is a given value. The function is defined as:
arccos(x) = y ⟺ cos(y) = x
where -1 ≤ x ≤ 1 and 0 ≤ y ≤ π
Domain and Range
- •Domain: [-1, 1] (all possible cosine values)
- •Range: [0, π] radians or [0°, 180°]
Key Properties
- •Strictly decreasing function
- •arccos(1) = 0° (minimum value)
- •arccos(-1) = 180° (maximum value)
- •arccos(0) = 90° (midpoint)
Negative Values Formula
arccos(-x) = π - arccos(x)
This symmetry property is useful for calculating negative inputs
Real-World Applications
Physics
Finding angles in vector calculations, determining directions in mechanics problems
Engineering
Calculating angles in structural analysis, robotics, and control systems
Computer Graphics
3D rotations, lighting calculations, and camera positioning algorithms