Calculate the angle (in degrees or radians) whose cosine is a given value.
Degrees (°)
Radians (rad)
Arccosine Result:
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Understanding the Inverse Cosine (Arccosine) Function
The inverse cosine, also known as arccosine or cos⁻¹, is a mathematical function that performs the inverse operation of the cosine function. While the cosine function takes an angle and returns the ratio of the adjacent side to the hypotenuse in a right-angled triangle (or a specific coordinate on the unit circle), the arccosine function takes this ratio (a value between -1 and 1) and returns the corresponding angle.
The Math Behind Arccosine
Mathematically, if:
cos(θ) = x
Then:
θ = arccos(x)
or
θ = cos⁻¹(x)
where:
θ (theta) represents the angle.
x represents the cosine value, which must be in the range [-1, 1].
The principal values for the arccosine function are typically defined as angles between 0 and 180 degrees (or 0 and π radians). This is because the cosine function is not strictly one-to-one over its entire domain; restricting the output range ensures a unique angle for each valid input value.
Units of Measurement
The output of the arccosine function can be expressed in two primary units:
Degrees: A full circle is 360 degrees. Angles are commonly expressed in degrees in many practical applications, especially in geometry and basic trigonometry.
Radians: A full circle is 2π radians. Radians are the standard unit for angles in higher mathematics, calculus, and physics, as they simplify many formulas.
Our calculator allows you to choose your preferred output unit.
How to Use the Calculator
Enter the Cosine Value. This number must be between -1 and 1, inclusive.
Select the desired Output Unit (Degrees or Radians).
Click the "Calculate Arccosine" button.
Use Cases for Arccosine
The inverse cosine function is fundamental in various fields:
Physics: Calculating angles in projectile motion, forces, and vectors. For example, determining the launch angle of a projectile given its horizontal range and initial velocity.
Engineering: Analyzing forces in structures, calculating angles in mechanical systems, and signal processing.
Computer Graphics: Determining the rotation needed for objects or the angle between vectors.
Navigation: Calculating bearings and positions.
Trigonometry Problems: Solving complex trigonometric equations and finding unknown angles in geometric shapes.
Example Calculation:
Let's find the angle whose cosine is 0.5.
Input Cosine Value: 0.5
Select Output Unit: Degrees
Calculation: arccos(0.5)
Result: 60°
If we choose Radians:
Input Cosine Value: 0.5
Select Output Unit: Radians
Calculation: arccos(0.5)
Result: 1.047... (which is π/3)
function calculateInverseCosine() {
var cosineValueInput = document.getElementById("cosineValue");
var angleUnit = document.getElementById("angleUnit").value;
var resultDiv = document.getElementById("result");
var errorMessageDiv = document.getElementById("error-message");
// Clear previous error messages
errorMessageDiv.textContent = "";
resultDiv.textContent = "–";
var cosineValue = parseFloat(cosineValueInput.value);
// Input validation
if (isNaN(cosineValue)) {
errorMessageDiv.textContent = "Error: Please enter a valid number for the cosine value.";
return;
}
if (cosineValue 1) {
errorMessageDiv.textContent = "Error: Cosine value must be between -1 and 1 (inclusive).";
return;
}
var angleInRadians = Math.acos(cosineValue);
var finalAngle;
var unitSymbol = "";
if (angleUnit === "degrees") {
finalAngle = angleInRadians * (180 / Math.PI);
unitSymbol = "°";
} else { // radians
finalAngle = angleInRadians;
unitSymbol = " rad";
}
// Format the result to a reasonable number of decimal places
resultDiv.textContent = finalAngle.toFixed(6) + unitSymbol;
}