Calculate the inverse cosine (arccos) of a ratio value. Maps ratio values between -1 and 1 back to angles.
The Arccos Calculator is an inverse trigonometric tool designed to compute the angle corresponding to a given cosine value. Arccosine (commonly written as arccos or cos⁻¹) is the inverse of the cosine function. Given an input ratio x, it calculates the angle θ such that cos(θ) = x. The output angle can be displayed in degrees or radians.
The domain of the arccosine function is restricted to the interval [−1, 1], representing the range of possible cosine values. Entering a value outside this interval will result in an error. To define arccos as a single-valued function, its range is restricted to principal values between 0° and 180° (or 0 and π radians). This differs from arcsin, which ranges from negative to positive 90 degrees.
Arccosine is widely used to calculate the interior angles of triangles using the Law of Cosines when all three side lengths are known (SSS case). It is also critical in three-dimensional vector analysis to find the angle between two vectors using their dot product. In astronomy, arccos is used to calculate the angular separation of stars. The arccos calculator provides instant, precise angle values, assisting users in mechanical engineering designs and algebra coursework.
Understanding Inverse Cosine (arccos / cos⁻¹) and the Unit Circle
In geometry, a trigonometric ratio relates the angles of a right triangle to its side lengths. When extended to all real coordinates, we use the **Unit Circle** (a circle with radius r = 1 centered at the origin).
- Sine (sin θ): represents the vertical projection coordinate (y-coordinate) of the intersection point.
- Cosine (cos θ): represents the horizontal projection coordinate (x-coordinate) of the intersection point.
- Tangent (tan θ): represents the ratio of vertical to horizontal change (y/x). Undefined when the cosine is 0 (90° and 270°).
- Pythagorean Identity: For any angle, the relationship sin²(θ) + cos²(θ) = 1 always holds.
Arccos Domain & Range: To build a unique inverse cosine, its range is restricted to [0°, 180°] ([0, π] radians). It yields the upper half of the unit circle.
How it Works & Formula
Calculates the inverse cosine, returning the angle whose cosine is the given value.
Practical Examples
arccos(0.5) = 60° (or π/3 radians).
Frequently Asked Questions
What is the range of arccos?
In degrees, the output ranges from 0° to 180°.