Inverse Cosine Calculator
Calculate arccos(x) - the inverse cosine function for values between -1 and 1
Calculate Inverse Cosine
Domain: [-1, 1]
Results
In Degrees
In Radians
Explanation
cos(60°) = 1/2, therefore arccos(1/2) = 60°
Special Values of Inverse Cosine
| Input (x) | arccos(x) in Degrees | arccos(x) in Radians | Exact Value |
|---|---|---|---|
| 1 | 0° | 0 | 0 |
| √3/2 | 30° | π/6 | π/6 ≈ 0.5236 |
| √2/2 | 45° | π/4 | π/4 ≈ 0.7854 |
| 1/2 | 60° | π/3 | π/3 ≈ 1.0472 |
| 0 | 90° | π/2 | π/2 ≈ 1.5708 |
| -1/2 | 120° | 2π/3 | 2π/3 ≈ 2.0944 |
| -√2/2 | 135° | 3π/4 | 3π/4 ≈ 2.3562 |
| -√3/2 | 150° | 5π/6 | 5π/6 ≈ 2.6180 |
| -1 | 180° | π | π ≈ 3.1416 |
Quick Reference
Properties
Understanding the Inverse Cosine Function
What is Inverse Cosine?
The inverse cosine function, denoted as arccos(x) or cos⁻¹(x), is the inverse function of the cosine. It takes a value between -1 and 1 and returns the angle whose cosine equals that value.
Key Properties
- •Domain: [-1, 1] - input must be between -1 and 1
- •Range: [0, π] radians or [0°, 180°]
- •The function is decreasing (as x increases, arccos(x) decreases)
- •arccos(x) + arcsin(x) = π/2 for all x in [-1, 1]
Applications
The inverse cosine function is used in trigonometry, engineering, physics, and computer graphics for finding angles when the cosine value is known.
Common Use Cases
- •Finding angles in triangles
- •Solving trigonometric equations
- •Vector calculations and dot products
- •Computer graphics and 3D rotations
Key Formulas
Basic Definition
Relationships
Step-by-Step Examples
Example 1: Find arccos(1/2)
Step 1: We need to find the angle whose cosine is 1/2
Step 2: From the unit circle, we know cos(60°) = 1/2
Step 3: Since 60° is in the range [0°, 180°], arccos(1/2) = 60°
Answer: arccos(1/2) = 60° = π/3 radians
Example 2: Find arccos(-√2/2)
Step 1: We need to find the angle whose cosine is -√2/2
Step 2: From the unit circle, we know cos(135°) = -√2/2
Step 3: Since 135° is in the range [0°, 180°], arccos(-√2/2) = 135°
Answer: arccos(-√2/2) = 135° = 3π/4 radians