Understanding Inverse Cosine (Arccos)
The inverse cosine function, written as arccos(x) or cos-1(x), returns the angle whose cosine is x. If cos(θ) = x, then arccos(x) = θ. The function is defined for inputs in the range [-1, 1] and produces outputs in the range [0°, 180°] or [0, π] radians.
Common Arccos Values
arccos(1) = 0°
The cosine of 0 degrees is 1, so the inverse cosine of 1 is 0 degrees.
arccos(√3/2) = 30°
One of the standard unit circle values from a 30-60-90 triangle.
arccos(√2/2) = 45°
The cosine of 45 degrees equals the square root of 2 divided by 2.
arccos(1/2) = 60°
A fundamental value from the 30-60-90 triangle on the unit circle.
arccos(0) = 90°
The cosine of 90 degrees is 0, giving arccos(0) = 90 degrees.
arccos(-1) = 180°
The cosine of 180 degrees is -1, the maximum output angle.
Domain and Range
The domain of arccos is [-1, 1], meaning you can only take the inverse cosine of values between -1 and 1 inclusive. The range is [0, π] radians or [0°, 180°], covering angles in the first and second quadrants only. This restriction ensures arccos is a proper function with exactly one output for each valid input.
Key Properties
- arccos(x) is a decreasing function: as x increases, the angle decreases.
- arccos(x) + arcsin(x) = π/2 for all x in [-1, 1].
- cos(arccos(x)) = x for all x in [-1, 1].
- arccos(-x) = π - arccos(x) (reflection property).
- The derivative: d/dx arccos(x) = -1/√(1 - x²).
Applications
- Navigation and GPS calculations for finding bearing angles.
- Computer graphics for calculating angles between vectors.
- Physics for determining angles of reflection and refraction.
- Engineering for structural analysis and force decomposition.