Arccos Calculator (Inverse Cosine)
Calculate the inverse cosine (arccos) of a value and understand its properties.
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Inverse Cosine (Arccos)
The inverse cosine function, denoted as arccos(x) or cos⁻¹(x), is the inverse of the cosine function.
Definition
If y = arccos(x), then x = cos(y) where:
- \[ -1 \leq x \leq 1 \]
- \[ 0 \leq y \leq \pi \] (in radians)
- \[ 0° \leq y \leq 180° \] (in degrees)
Properties
- Domain: \[ [-1, 1] \]
- Range: \[ [0, \pi] \] (radians) or \[ [0°, 180°] \] (degrees)
- Monotonic: Decreasing function
- Symmetry: \[ \arccos(-x) = \pi - \arccos(x) \]
Common Values
- \[ \arccos(1) = 0 \]
- \[ \arccos(0) = \frac{\pi}{2} \] (90°)
- \[ \arccos(-1) = \pi \] (180°)
- \[ \arccos(\frac{1}{2}) = \frac{\pi}{3} \] (60°)
- \[ \arccos(-\frac{1}{2}) = \frac{2\pi}{3} \] (120°)
Derivative
\[ \frac{d}{dx}\arccos(x) = -\frac{1}{\sqrt{1-x^2}} \]
Applications
- Trigonometry
- Calculus
- Physics (wave motion)
- Engineering (signal processing)
- Computer graphics
Relationship with Other Functions
- \[ \arccos(x) = \frac{\pi}{2} - \arcsin(x) \]
- \[ \arccos(x) = \arctan(\frac{\sqrt{1-x^2}}{x}) \]
- \[ \arccos(x) = \arccot(\frac{x}{\sqrt{1-x^2}}) \]