Arccos Calculator
Calculate arccos instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Free to use with no signup required.
Formula
arccos(x) = angle where cos(angle) = x, for x in [-1, 1]
The inverse cosine function returns the angle (in radians from 0 to pi, or degrees from 0 to 180) whose cosine equals the input value x. The input must be between -1 and 1 inclusive.
Worked Examples
Example 1: Finding an Angle from a Cosine Value
Problem: The cosine of an angle is 0.7071. Find the angle in degrees and radians.
Solution: arccos(0.7071) = 45.0004 degrees (approximately 45 degrees)\nIn radians: 45 x pi/180 = 0.7854 radians = pi/4\n\n0.7071 is approximately sqrt(2)/2 = 1/sqrt(2)\nThis is the cosine of 45 degrees in a 45-45-90 triangle\n\nVerification: cos(45 degrees) = cos(pi/4) = sqrt(2)/2 = 0.7071\nsin(45 degrees) = sqrt(2)/2 = 0.7071\ntan(45 degrees) = 1.0000
Result: arccos(0.7071) = 45 degrees = pi/4 radians
Example 2: Angle Between Two Vectors
Problem: Find the angle between vectors A = (3, 4) and B = (1, 0).
Solution: A dot B = 3(1) + 4(0) = 3\n|A| = sqrt(9 + 16) = sqrt(25) = 5\n|B| = sqrt(1 + 0) = 1\n\ncos(theta) = (A dot B) / (|A| x |B|) = 3 / (5 x 1) = 0.6\ntheta = arccos(0.6) = 53.1301 degrees\n\nIn radians: 53.1301 x pi/180 = 0.9273 radians
Result: Angle between vectors = 53.13 degrees = 0.9273 radians
Frequently Asked Questions
What is arccos (inverse cosine)?
Arccos, also written as cos^(-1) or acos, is the inverse function of cosine. Given a value x between -1 and 1, arccos(x) returns the angle whose cosine equals x. The function is defined for inputs in the range [-1, 1] and produces output angles in the range [0, 180] degrees or [0, pi] radians. This restricted output range is called the principal value and ensures the function gives exactly one answer for each input. For example, arccos(0.5) = 60 degrees because cos(60 degrees) = 0.5. The arccos function is essential in trigonometry, physics, vector mathematics, and computer graphics for finding angles from known cosine ratios.
What is the domain and range of the arccos function?
The domain of arccos is the closed interval [-1, 1], meaning you can only take the arccos of values between -1 and 1 inclusive. Any input outside this range is undefined because cosine values never exceed 1 or go below -1. The range (output) of arccos is [0, pi] radians or equivalently [0, 180] degrees. This means arccos always returns an angle between 0 and 180 degrees. At the boundary values: arccos(1) = 0 degrees, arccos(0) = 90 degrees, and arccos(-1) = 180 degrees. The range restriction is necessary because cosine is not one-to-one over its entire domain, so the inverse must be restricted to a single period where cosine is monotonically decreasing.
How is arccos different from arcsin and arctan?
While all three are inverse trigonometric functions, they differ in their domains, ranges, and the triangles they solve. Arccos has domain [-1, 1] and range [0, pi] (0 to 180 degrees). Arcsin has domain [-1, 1] and range [-pi/2, pi/2] (-90 to 90 degrees). Arctan has domain (-infinity, infinity) and range (-pi/2, pi/2) (-90 to 90 degrees). There is a key complementary relationship: arccos(x) + arcsin(x) = pi/2 (90 degrees) for all x in [-1, 1]. This means arccos(0.5) = 60 degrees and arcsin(0.5) = 30 degrees, and they sum to 90. Arccos is particularly useful when you know the adjacent side and hypotenuse of a right triangle, while arcsin is used when you know the opposite side and hypotenuse.
What are the common arccos values to memorize?
The most important arccos values come from special angles used in trigonometry. arccos(1) = 0 degrees, arccos(sqrt(3)/2) = 30 degrees, arccos(sqrt(2)/2) = 45 degrees, arccos(1/2) = 60 degrees, arccos(0) = 90 degrees, arccos(-1/2) = 120 degrees, arccos(-sqrt(2)/2) = 135 degrees, arccos(-sqrt(3)/2) = 150 degrees, and arccos(-1) = 180 degrees. These values correspond to the special angles on the unit circle and appear constantly in mathematics, physics, and engineering. Memorizing these values helps you quickly solve trigonometric equations and verify calculator results. Notice the symmetry: arccos(-x) = 180 degrees minus arccos(x).
How is arccos used in vector mathematics?
In vector mathematics, arccos is the standard method for finding the angle between two vectors. The dot product formula states that A dot B = |A| times |B| times cos(theta), where theta is the angle between vectors A and B. Rearranging gives theta = arccos(A dot B / (|A| times |B|)). For example, if vectors A = (1, 0) and B = (1, 1), then A dot B = 1, |A| = 1, |B| = sqrt(2), so theta = arccos(1/sqrt(2)) = 45 degrees. This application is crucial in computer graphics for lighting calculations (angle between surface normal and light direction), in physics for calculating work (force dot displacement), and in machine learning for cosine similarity between feature vectors.
What is the derivative of arccos?
The derivative of arccos(x) with respect to x is -1 / sqrt(1 - x^2). Note the negative sign, which distinguishes it from the derivative of arcsin(x) which is positive: +1 / sqrt(1 - x^2). The derivative is undefined at x = -1 and x = 1 because the denominator becomes zero at these points, corresponding to the endpoints of the domain where the arccos curve has vertical tangent lines. The negative derivative confirms that arccos is a strictly decreasing function: as the input x increases from -1 to 1, the output angle decreases from 180 to 0 degrees. This derivative formula is essential in calculus for integrating expressions involving arccos and for related rates problems in physics and engineering.