Last updated: July 17, 2026
Cos⁻¹ Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The cos⁻¹ (arccos) calculator computes the inverse cosine of any value x in [−1, 1], returning the unique angle θ in [0, π] whose cosine equals x. Results appear in both radians and degrees. The nine standard special angles are recognised exactly. The function is strictly decreasing: arccos(1) = 0 and arccos(−1) = π.
To find the inverse cosine of x, enter x between −1 and 1. The calculator returns arccos(x) — the angle whose cosine is x — in radians and degrees. For example, arccos(0.5) equals π/3 radians or 60 degrees.
Key Takeaways
- arccos(x) finds the unique angle θ in [0, π] whose cosine equals x.
- The domain is strictly [−1, 1]; inputs outside this range yield an undefined result.
- The function is strictly decreasing: larger x gives a smaller angle.
- Nine special angles have exact symbolic arccos values, e.g. arccos(0.5) = π/3 = 60°.
- The identity arccos(x) + arcsin(x) = π/2 links the two main inverse trig functions.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
arccos(x) = θ such that cos(θ) = x, where θ ∈ [0, π]
Where:
- x=Cosine value (input)(dimensionless)
- θ=Angle (arccos result)(radians or degrees)
Worked Examples
Standard Angle: cos⁻¹(1/2) = 60°
Find the angle whose cosine equals 0.5. This is one of the most common trigonometric special values.
- 1Identify the input: x = 0.5 (cosine value)
- 2Domain check: −1 ≤ 0.5 ≤ 1 ✓ — input is valid
- 3Apply arccos: arccos(0.5) = ?
- 4Recall that cos(π/3) = cos(60°) = 0.5
- 5Therefore arccos(0.5) = π/3 radians = 60°
Right Angle: cos⁻¹(0) = 90°
Find the angle whose cosine equals zero. Cosine is zero at a right angle, the midpoint of the unit circle's top semicircle.
- 1Identify the input: x = 0
- 2Domain check: −1 ≤ 0 ≤ 1 ✓
- 3Apply arccos: arccos(0) = ?
- 4Recall that cos(π/2) = 0
- 5Result: arccos(0) = π/2 ≈ 1.5708 radians (= 90°)
Obtuse Angle: cos⁻¹(−1/2) = 120°
Find the angle whose cosine equals −0.5. Negative cosine values always correspond to obtuse angles in (90°, 180°).
- 1Identify the input: x = −0.5 (negative cosine value)
- 2Domain check: −1 ≤ −0.5 ≤ 1 ✓
- 3A negative cosine means the angle is in (π/2, π), i.e. obtuse
- 4Recall that cos(2π/3) = cos(120°) = −0.5
- 5Result: arccos(−0.5) = 2π/3 radians = 120°
Introduction
The inverse cosine function, written as cos⁻¹(x) or arccos(x), answers the question: what angle has a cosine of x? Given any value x in the domain [−1, 1], it returns a unique angle θ in [0, π] radians (equivalently [0°, 180°]). This calculator computes arccos instantly and shows the result in both radians and degrees, covering all nine standard special angles as exact symbolic values.
What Is the Inverse Cosine (arccos)?
The cosine function maps angles to values in −1, 1]. The inverse cosine reverses this: given a cosine value x, arccos(x) returns the angle θ in [0, π] whose cosine is x. Because cosine is not globally one-to-one, a restricted domain [0, π] is chosen so that arccos is a proper function. Mathematically: if cos(θ) = x, then arccos(x) = θ. This gives only the principal value in [0, π]. For related inverse-trig tools see the [arccos calculator and the inverse sine calculator.
Domain, Range, and Key Properties
Domain: −1 ≤ x ≤ 1. Any cosine value outside this interval is undefined for real angles. Range: [0, π] radians = [0°, 180°]. The output is always a non-negative angle no greater than π. arccos is strictly decreasing: as x increases from −1 to 1, the angle decreases from π to 0. The reflection identity arccos(−x) = π − arccos(x) means arccos(−0.5) = π − arccos(0.5) = π − π/3 = 2π/3 = 120°.
Special Angle Values
Nine special angles cover the standard unit-circle positions: arccos(1)=0, arccos(√3/2)=π/6 (30°), arccos(√2/2)=π/4 (45°), arccos(1/2)=π/3 (60°), arccos(0)=π/2 (90°), arccos(−1/2)=2π/3 (120°), arccos(−√2/2)=3π/4 (135°), arccos(−√3/2)=5π/6 (150°), arccos(−1)=π (180°). For arbitrary values consult NIST DLMF §4.23 for series expansions and additional identities.
Relationship to Other Inverse Trig Functions
Key identities connecting arccos to other inverses: arccos(x) + arcsin(x) = π/2 for all x ∈ −1, 1], so if you know arcsin you can get arccos instantly. Also arccos(x) = π/2 − arctan(x / √(1−x²)). Use the [inverse tangent calculator when recovering an angle from a tangent value. Wolfram MathWorld covers the full family of inverse trigonometric identities.
Radians vs. Degrees
arccos always returns radians internally (as all standard math libraries do). Convert: radians to degrees by multiplying by 180/π ≈ 57.2958; degrees to radians by multiplying by π/180. Radians are required for calculus: d/dx[arccos x] = −1/√(1−x²). This calculator displays both units side by side.
Applications in Geometry and Engineering
arccos is used in: triangle geometry via the Law of Cosines A = arccos((b² + c² − a²)/(2bc)) to find any angle from three side lengths; vector angle computation θ = arccos((u·v)/(|u||v|)); computer graphics and inverse kinematics to recover joint angles; signal processing for phase-difference extraction; and geodesy where great-circle distance formulas use arccos after the haversine step. See Encyclopaedia Britannica on trigonometry for further context.
Quick Reference Card
arccos Quick Reference
Quick reference • Cos⁻¹ Calculator
arccos(x) = θ where cos(θ) = x, θ ∈ [0, π]Valid range: −1 ≤ x ≤ 1
Common Values
⚠ Watch Out
- •Input must satisfy −1 ≤ x ≤ 1; otherwise the result is undefined (NaN).
- •cos⁻¹(x) is NOT the same as 1/cos(x) — that is the secant function.
- •arccos returns only the principal value in [0, π]; infinitely many angles share any given cosine.
- •When arccos receives a floating-point value like 1.0000000002, clamp it to 1 first to avoid NaN.
Pro Tips
- →Use the identity arccos(x) = π/2 − arcsin(x) to convert quickly between the two inverse functions.
- →In the Law of Cosines, clamp the argument to [−1, 1] before calling arccos to guard against floating-point rounding errors.
- →Prefer Math.atan2 over arccos when computing vector angles in code; it handles all four quadrants cleanly.
- →For high-precision work near x = 1, use arccos(x) = 2·arctan(√((1−x)/(1+x))) which is numerically stable.
FAQs
What is the domain of arccos?
The domain is [−1, 1]. Cosine values are always between −1 and 1 for real angles, so any input outside this interval has no real arccos result.
What is the range of arccos?
The range is [0, π] radians, which equals [0°, 180°]. The function returns only the principal value in this interval to ensure a unique output for each input.
Is cos⁻¹(x) the same as 1/cos(x)?
No. cos⁻¹(x) denotes the inverse cosine function (arccos), not the reciprocal. The reciprocal of cosine is the secant: sec(x) = 1/cos(x). The superscript −1 in cos⁻¹ denotes the inverse function.
Why is arccos a decreasing function?
Because cosine itself decreases on [0, π]: as the angle grows from 0 to π, the cosine falls from 1 to −1. Inverting this relationship reverses the direction: a larger cosine value corresponds to a smaller angle.
How do I convert the arccos result from radians to degrees?
Multiply the radian value by 180/π. For example, arccos(0.5) = π/3 radians × (180/π) = 60°. This calculator displays both units automatically.
What is arccos(cos(x)) for an angle outside [0, π]?
arccos(cos(x)) returns the principal-value angle in [0, π] that shares the same cosine. For example, cos(240°) = −0.5, so arccos(cos(240°)) = arccos(−0.5) = 120°, not 240°.