Arccos Calculator

What is the arccos function?

The arccosine, written arccos(x), acos(x) or cos⁻¹(x), is the inverse of the cosine. The cosine takes an angle and returns a ratio; the arccosine takes that ratio and gives the angle back.

if cos(θ) = x, then θ = arccos(x)

Many different angles share the same cosine, so the inverse needs a convention to be a function. By definition arccos returns the one angle between 0° and 180° (0 and π radians). That is why arccos(0.5) is 60° and not −60° or 300°, even though all three have a cosine of 0.5.

cos⁻¹(x) means the inverse function, not the reciprocal. The reciprocal 1 ÷ cos(x) is the secant.

Domain and range

The cosine of a real angle always lies between −1 and 1, so the arccosine exists only for inputs in that interval. Asking for arccos(1.5) has no real answer, and the calculator will tell you the input is out of range.

• Domain: −1 ≤ x ≤ 1.
• Range: 0 ≤ arccos(x) ≤ π radians, which is 0° to 180°.
• arccos(1) = 0 and arccos(−1) = π (180°).
• The function is decreasing: as x grows, the angle gets smaller.

How to use the calculator

Enter the cosine as a decimal (0.5, −0.3) or as a fraction (1/3, −3/4). A fraction is divided out before the angle is found, and the heading of each result shows the value that was used. You get the angle twice: once in degrees and once in radians, each to up to six decimal places.

To go the other way, from an angle to its cosine, use the cosine calculator. For the companions of this function, see the arcsine calculator and the arctangent calculator.

Common arccos values

Useful identities

cos(arccos x) = x for −1 ≤ x ≤ 1

arccos(−x) = π − arccos(x)

arccos(x) + arcsin(x) = π ÷ 2

arccos(x) = arctan(√(1 − x²) ÷ x) for x > 0

The second identity is the mirror symmetry of the cosine: arccos(0.4) is about 66.42°, so arccos(−0.4) is 180° − 66.42° = 113.58°. The third links arccos to the arcsine, since the two angles of a right triangle's acute corners add up to 90°.

Worked examples

Right triangle. A right triangle has a hypotenuse of 5 and a side next to the angle of 3. The cosine of that angle is adjacent ÷ hypotenuse = 3 ÷ 5 = 0.6, so the angle is arccos(0.6) ≈ 53.130102°, or 0.927295 radians.

Ladder. A 4 m ladder has its foot 1.2 m from a wall. The angle between the ladder and the ground satisfies cos θ = 1.2 ÷ 4 = 0.3, so θ = arccos(0.3) ≈ 72.542397°.

Obtuse angle. For x = −0.3 the answer is arccos(−0.3) ≈ 107.457603°, because 180° − 72.542397° = 107.457603°. A negative cosine always gives an angle above 90°.

Tips and common mistakes

• Check the mode of your own calculator. Degrees and radians give completely different numbers for the same input.
• Only one angle is returned. If your problem allows other solutions, such as the reflex angle 360° − θ, add them yourself.
• Rounding can push a value just outside the interval. A cosine of 1.0000001 caused by rounding should be clamped to 1 before taking the arccosine.
• When the cosine comes from side lengths, divide the sides first. A ratio that lands outside −1 to 1 means the lengths cannot form that triangle, so recheck the measurements.