The ArcCosine (Inverse Cosine) Function
ArcCosine returns the angle whose cosine equals the given value. arccos(x) = θ where cos(θ) = x. Domain: [-1, 1], Range: [0, π] — outputs angles in Q1 and Q2.
Why This Mathematical Concept Matters
Why: ArcCosine is used for the angle between vectors (cosine similarity), 3D graphics, and solving equations like cos(θ) = x.
How: arccos(x) returns the unique angle in [0, π] such that cos(θ) = x. Unlike arcsin, the range covers Q1 and Q2 — useful for angles from 0° to 180°.
- ●arccos(x) + arcsin(x) = π/2 for all x ∈ [-1, 1] — complementary angles.
- ●arccos(x) gives the angle between two vectors: θ = arccos(a·b / (|a||b|)).
- ●d/dx[arccos(x)] = -1/√(1-x²); the negative sign reflects the decreasing nature of cosine.
Examples — Click to Load
Inverse Trig Value Breakdown
All Related Inverse Values
arccos vs arcsin (Complementary)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
arccos gives the angle between vectors — used in cosine similarity and 3D graphics.
— Khan Academy
arccos(-x) = π - arccos(x) — reflection property; arccos is neither odd nor even.
— Paul's Notes
Key Takeaways
- • arccos(x) returns the angle whose cosine is x. Domain: [-1, 1], Range: [0, π] (principal value)
- • arccos(0)=90°, arccos(0.5)=60°, arccos(√2/2)=45°, arccos(1)=0°, arccos(-1)=180°. arccos(-x) = π - arccos(x)
- • arccos(x) + arcsin(x) = π/2 — complementary angles for x ∈ [-1, 1]; used in dot product angle computation
- • cos(arccos(x)) = x for x ∈ [-1, 1]. arccos(cos(θ)) = θ only when θ ∈ [0, π]
- • Range [0, π] is chosen so arccos is a true function; essential for 3D graphics and vector angle calculations
Did You Know?
How ArcCosine Works
arccos(x) finds the unique angle θ in [0, π] such that cos(θ) = x. Since cosine is not one-to-one on its full domain, we restrict the range to obtain a well-defined inverse.
Principal Value
The principal value of arccos is the angle in [0°, 180°]. For cos(θ) = 0.5, both 60° and 300° work, but arccos(0.5) = 60° by convention.
arccos(cos(x)) Considerations
arccos(cos(x)) = x only when x ∈ [0, π]. For x = 5π/3, cos(5π/3)=0.5 but arccos(0.5)=π/3. The inverse "undoes" only within the principal range.
Complementary Identity & Dot Product
arccos(x) + arcsin(x) = π/2 for all x ∈ [-1, 1]. In vector geometry, the angle between vectors a and b is θ = arccos(a·b / (|a||b|)), used extensively in 3D graphics.
Expert Tips
Memorize Key Values
arccos(1)=0°, arccos(√3/2)=30°, arccos(√2/2)=45°, arccos(0.5)=60°, arccos(0)=90°, arccos(-1)=180°. Use the Unit Circle Calculator to visualize.
Watch Domain Restrictions
arccos is undefined for |x| > 1. For dot product angles, ensure vectors are normalized. See this calculator for real inputs.
Use arcsin for Complementary
If you need the angle in [-π/2, π/2], use arcsin(x) instead. arccos(x) + arcsin(x) = 90° always. Try the ArcSine Calculator.
Verify with cos(arccos(x))
Always check: cos(arccos(x)) = x. This verification catches domain errors. Use the Cosine Calculator to confirm.
Inverse Trig Calculator Comparison
| Feature | ArcSine | ArcCosine | ArcTangent |
|---|---|---|---|
| Domain | [-1, 1] | [-1, 1] | All reals |
| Range | [-π/2, π/2] | [0, π] | (-π/2, π/2) |
| Principal value | Yes | Yes | Yes |
| Odd/Even | Odd | Neither | Odd |
| Complement | arccos(x) | arcsin(x) | arccot(x) |
| Undefined for | |x| > 1 | |x| > 1 | Never |
| Common use | Opp/hyp ratio | Adj/hyp, dot product | Slope angle |
| 3D graphics | Less common | Vector angles | atan2 preferred |
Frequently Asked Questions
Why is arccos only defined for [-1, 1]?
Because cosine only outputs values between -1 and 1. There is no real angle whose cosine equals 2 or -1.5. The domain of an inverse function equals the range of the original function.
What is arccos(-0.5)?
120° (or 2π/3 rad). arccos(-x) = π - arccos(x). The negative input gives an angle in the second quadrant, between 90° and 180°.
Why does arccos(cos(240°)) = 120°?
cos(240°) = -0.5, and arccos(-0.5) returns the principal value 120°, not 240°. The inverse only "undoes" within the range [0°, 180°].
What is arccos(x) + arcsin(x)?
Always π/2 (90°) for x ∈ [-1, 1]. They are complementary angles — the two acute angles in a right triangle sum to 90°.
How is arccos used in dot product angle computation?
For vectors a and b, the angle between them is θ = arccos(a·b / (|a||b|)). This formula is fundamental in 3D graphics, physics, and machine learning.
How is arccos used in calculus?
d/dx[arccos(x)] = -1/√(1-x²). The integral ∫ -dx/√(1-x²) = arccos(x) + C. Essential for trigonometric substitution.
Why is the range [0, π] chosen for arccos?
Cosine is strictly decreasing on [0, π], so the inverse is unique. This interval covers all possible cosine values and is the standard convention in math and programming.
Can arccos output angles in Q3 or Q4?
No. The principal range is [0, π], which covers only Q1 and Q2. For angles in Q3 or Q4, use arcsin or adjust with 2π - arccos(x).
ArcCosine by the Numbers
Official & Educational Sources
Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices, 3D graphics), always verify with certified computational tools. Inputs outside [-1, 1] are undefined for real arccos.