Arccos Calculator (Inverse Cosine)
Calculate arccos(x) - the inverse cosine function
Arccos is only defined for values between -1 and 1
Common Arccos Values
arccos(1)
0°
0
arccos(√3/2 ≈ 0.866)
30°
π/6
arccos(√2/2 ≈ 0.707)
45°
π/4
arccos(0.5 (1/2))
60°
π/3
arccos(0)
90°
π/2
arccos(-0.5)
120°
2π/3
arccos(-√2/2)
135°
3π/4
arccos(-√3/2)
150°
5π/6
arccos(-1)
180°
π
How to Use This Calculator
Enter Cosine Value
Input the cosine value (x) you want to find the angle for. This value must be between -1 and 1.
Calculate
Click the "Calculate Arccos" button to get the angle in both degrees and radians.
Review Results
View the angle result and verify it by checking that cos(angle) equals your input value.
Formula
arccos(x) = θ, where cos(θ) = x
Principal value range: 0° to 180° (0 to π radians)
Definition:
The arccos function (also written as cos⁻¹) is the inverse of the cosine function. Given a cosine value x, arccos(x) returns the angle θ such that cos(θ) = x.
Domain and Range:
- Domain: -1 ≤ x ≤ 1 (only these values are valid inputs)
- Range: 0° ≤ arccos(x) ≤ 180° (0 ≤ arccos(x) ≤ π radians)
- The principal value is always in the first or second quadrant
Relationship to Cosine:
cos(arccos(x)) = x, for -1 ≤ x ≤ 1
Key Properties:
- arccos(-x) = 180° - arccos(x)
- arccos(1) = 0°
- arccos(0) = 90°
- arccos(-1) = 180°
About Arccos Calculator
The Arccos Calculator (Inverse Cosine Calculator) is a powerful tool that finds the angle whose cosine equals a given value. Arccos is the inverse function of cosine, meaning it "undoes" the cosine operation. This calculator helps you quickly determine angles from cosine values, which is essential in trigonometry, geometry, and various engineering applications.
What is Arccos?
Arccos, written as arccos(x) or cos⁻¹(x), is the inverse trigonometric function of cosine. Given a cosine value x between -1 and 1, arccos(x) returns the angle (in the principal range of 0° to 180°) whose cosine is x. It answers the question: "What angle has a cosine of x?"
When to Use This Calculator
- Triangle Problems: Find angles when you know side ratios (adjacent/hypotenuse)
- Vector Mathematics: Calculate angles between vectors using dot product formulas
- Physics: Determine angles in force, velocity, and acceleration calculations
- Engineering: Solve problems involving angles in mechanical systems
- Navigation: Calculate bearings and directions from coordinate data
- Signal Processing: Analyze phase angles and waveform relationships
- Verification: Check solutions to trigonometric equations
Why Use Our Calculator?
- ✅ Instant Results: Get accurate angle values immediately in both degrees and radians
- ✅ Input Validation: Automatically checks that inputs are in valid range (-1 to 1)
- ✅ Verification: Shows that cos(result) equals your input to confirm accuracy
- ✅ Easy to Use: Simple interface requiring only one input value
- ✅ 100% Free: No registration or payment required
- ✅ Mobile Friendly: Works perfectly on all devices
- ✅ Educational: Helps understand the relationship between cosine and arccos
Important Notes
- Restricted Domain: Arccos only accepts values between -1 and 1 (the range of cosine)
- Principal Value: The calculator returns the principal value between 0° and 180°
- Multiple Solutions: For any cosine value, there are infinitely many angles with that cosine (due to periodicity), but arccos returns only the principal value
- Quadrants: Arccos returns angles in quadrants I (0°-90°) and II (90°-180°)
Frequently Asked Questions
What is arccos(1)?
arccos(1) = 0°. This is because cos(0°) = 1. The arccos of 1 gives the angle with maximum cosine value.
What is arccos(0)?
arccos(0) = 90° (or π/2 radians). This is because cos(90°) = 0. The arccos of 0 gives the angle at the top of the unit circle.
Why is arccos only defined for values between -1 and 1?
Arccos is the inverse of cosine. Since cosine only produces values between -1 and 1 (as cosine represents the x-coordinate on a unit circle), arccos can only "reverse" values in that range. If you input a value outside -1 to 1, there is no real angle with that cosine value.
What's the difference between arccos and cos⁻¹?
They are the same function! Both arccos(x) and cos⁻¹(x) represent the inverse cosine function. The notation cos⁻¹ is more common in some contexts, while arccos is clearer in distinguishing it from 1/cos(x), which is the reciprocal, not the inverse.
Can arccos return negative angles?
No, the principal value of arccos is always between 0° and 180° (0 and π radians). However, in general, there are infinitely many angles with the same cosine value, but arccos specifically returns the principal value in the range [0°, 180°].
Why does arccos return angles up to 180° but not higher?
The range of arccos is restricted to [0°, 180°] to make it a function (each input has exactly one output). While angles like 300° also have the same cosine as some angles in [0°, 180°], arccos always returns the principal value in the first or second quadrant.
How is arccos related to other inverse trigonometric functions?
All inverse trigonometric functions work similarly - they reverse their respective functions. For example: arccos(x) finds the angle with cosine x, arcsin(x) finds the angle with sine x, and arctan(x) finds the angle with tangent x. Each has its own domain and range restrictions.