Cos Inverse
Calculate inverse cosine (arccos) of any value between -1 and 1. Get the angle in degrees and radians with an interactive arccos function graph.
About This Calculator
The inverse cosine function, denoted cos⁻¹(x) or arccos(x), returns the angle whose cosine equals the input value x. It is the inverse of the cosine function. The principal value lies between 0 deg and 180 deg (0 to pi radians), covering the upper half of the unit circle.
The mathematical relationship is: theta = arccos(x) where cos(theta) = x and x must satisfy -1 <= x <= 1. This calculator provides both degree and radian outputs simultaneously, essential for students studying trigonometry, calculus, and advanced mathematics.
Inverse cosine is critical in fields like computer graphics (camera angles, lighting calculations), physics (work and energy calculations involving dot products), engineering (stress analysis at angles), navigation (great-circle distance calculations), and machine learning (cosine similarity for vector comparisons).
Frequently Asked Questions
What is cos⁻¹ or inverse cosine?
Cos inverse (arccos) is the inverse function of cosine. If cos(theta) = x, then arccos(x) = theta. It returns the angle whose cosine equals the given number. The output range is 0 deg to 180 deg (0 to pi radians).
What values can I input for cos inverse?
The inverse cosine function only accepts values between -1 and 1 (inclusive). This is because the cosine of any real angle always falls within the range [-1, 1]. Values outside this domain produce complex (non-real) results.
How does arccos differ from arcsin?
Arccos returns angles in the range 0 deg to 180 deg (0 to pi rad), while arcsin returns angles from -90 deg to 90 deg (-pi/2 to pi/2 rad). Both accept inputs in [-1, 1], but they cover different output quadrants. For example, arccos(0) = 90 deg but arcsin(0) = 0 deg.
Where is inverse cosine used in real life?
Inverse cosine is used in physics for calculating angles from dot products, in 3D graphics for determining viewing angles, in robotics for joint angle calculation, in astronomy for celestial coordinate conversion, and in signal processing for phase detection.
What is the formula for arccos?
Arccos(x) can be expressed as: arccos(x) = pi/2 - arcsin(x). It can also be computed using the series expansion. Common values: arccos(0) = 90 deg, arccos(1) = 0 deg, arccos(-1) = 180 deg, arccos(0.5) = 60 deg.
Is this cos inverse calculator free?
Yes, all calculators on Calculy are completely free to use. No registration or download required.