Find the inverse cosine (arccos) value for any input between -1 and 1. Get results in degrees and radians with step-by-step solutions.
A ladder of length 5 meters reaches a wall with its base 2.5 meters from the wall.
Formula: θ = arccos(adjacent / hypotenuse) = arccos(2.5/5)
arccos(0.5) ≈ 60°
Arccos gives the angle from the adjacent side and hypotenuse, useful when the base distance and ladder length are known.
A roof has a horizontal span of 6 meters and a rafter length (hypotenuse) of 7.21 meters.
Formula: θ = arccos(span / rafter) = arccos(6/7.21)
arccos(0.832) ≈ 33.69°
The roof pitch angle can be found using the horizontal run and the rafter length with the arccosine function.
A projectile is launched at a speed of 50 m/s and reaches a horizontal range of 220.6 meters.
Formula: θ = (1/2) × arccos(gR/v²) where g = 9.81, R = range, v = velocity
arccos(0.866) ≈ 30.00°
Arccos is used in physics to determine launch angles from range and initial velocity.
A radio tower is 50 meters tall. The horizontal distance from the observer to the tower is 100 meters, and the line-of-sight distance is 111.8 meters.
Formula: θ = arccos(adjacent / hypotenuse) = arccos(100/111.8)
arccos(0.8944) ≈ 26.57°
Arccos calculates the elevation angle when the horizontal distance and slant distance are known.
The arccosine function (arccos or cos⁻¹) is the inverse of the cosine function. For a given value x, arccos(x) returns the angle θ whose cosine is x. In a right triangle, this is the angle whose adjacent/hypotenuse ratio equals the input value.
arccos(x) returns an angle between 0° and 180° (inclusive). For x > 0, the angle is in Quadrant I. For x < 0, the angle is in Quadrant II.
As x increases from -1 to 1, arccos(x) decreases from 180° to 0°. The function is strictly decreasing.
arccos(x) is only defined for x in [-1, 1]. Values outside this range are not valid for real-number arccosine, since cosine values always lie within [-1, 1].
For any x in [-1, 1], arccos(x) + arcsin(x) = 90° (π/2). This relationship is fundamental in trigonometry.
The arccosine function (abbreviated arccos, cos⁻¹, or acos) is the inverse of the cosine function. While the cosine function takes an angle and returns a ratio, the arccosine function takes a ratio and returns the corresponding angle. It answers the question: "What angle θ has a cosine equal to x?"
In a right triangle, the cosine of an angle θ is the ratio of the adjacent side to the hypotenuse: cos(θ) = adjacent/hypotenuse. Therefore, arccos(adjacent/hypotenuse) = θ. This makes arccosine essential for determining angles when you know the side lengths of a triangle, particularly when you have the adjacent side and the hypotenuse.
The arccosine function has a domain of [-1, 1] and a range of [0, π] radians (or 0° to 180°). Any cosine value must fall within [-1, 1], and the arccosine will always return a principal value within this range. Unlike arcsine which is odd, arccosine is a decreasing function — as x increases, arccos(x) decreases.
Calculating arccosine values can be done in several ways. The most direct method is using the inverse trigonometric functions available on scientific calculators or in programming languages (acos, Math.acos, or arccos). For common values, the arccosine can be derived from known cosine ratios.
Method 1 — Direct Calculation: Enter the cosine value x (must be between -1 and 1) into a calculator and press the cos⁻¹ or arccos function. For example, arccos(0.5) = 60° because cos(60°) = 0.5.
Method 2 — Ratio Method: If you know the adjacent side and hypotenuse of a right triangle, divide the adjacent by the hypotenuse to get x, then apply the arccos function. For example, a ladder with its base 1 meter from the wall and a length of 2 meters has an angle of arccos(1/2) = arccos(0.5) = 60°.
Method 3 — Special Values: For the common angles (0°, 30°, 45°, 60°, 90°), the cosine values are exact: cos(0°) = 1, cos(30°) = √3/2 ≈ 0.8660, cos(45°) = √2/2 ≈ 0.7071, cos(60°) = 0.5, cos(90°) = 0. The arccosine of these values gives the corresponding angle.
These exact values are the building blocks of inverse trigonometry:
The arccosine function may seem like a purely mathematical concept, but it has numerous practical applications that affect our everyday lives. Understanding arccosine helps in fields ranging from construction to robotics, from navigation to game development.
Safety guidelines recommend specific angles for ladders and ramps. When you measure the base distance and the ladder/ramp length, arccosine converts these measurements into the angle of incline.
In sports analytics, arccosine helps calculate launch angles for projectiles. When you know the horizontal range and launch speed, arccosine determines the optimum angle for maximum distance.
In game development, arccosine is used in 3D environments for calculating horizontal camera angles, field-of-view adjustments, and character aiming mechanics involving azimuth.
Pilots use arccosine to calculate drift angles and bearing corrections. Given the groundspeed and airspeed vectors, arccosine provides the angle needed to correct course.
⚠️ Important Note: This Arccosine Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering, construction, navigation, or medical devices. Always consult a qualified professional for decisions involving trigonometric calculations in high-stakes contexts.