Arccos of −1/2

Arcsin, Arccos, Arctan, Arccot
Function
x

What is arccos(−1/2)?

Answer

arccos(−1/2) =
120° = 2π/3

Range: arccos x is an angle from 0° to 180°, so the answer is unique.

−1/2 = −0.5.

arccos(−1/2) is the angle α in [0°, 180°] with cos α = −1/2.

arccos(−1/2) = 180° − arccos(1/2) = 180° − 60° = 120°

cos 120° = −1/2 ✓

xyOP

α = arccos(−1/2) = 120°

P(cos 120°, sin 120°) = (−1/2, √3/2)

P lies on the line x = −1/2 in the upper half of the circle, where sin α ≥ 0; highlighted: cos α = −1/2.

All four inverse functions of −1/2
FunctionDegreesRadians
arcsin(−1/2)−30°−π/6
arccos(−1/2)120°2π/3
arctan(−1/2)≈ −26.5651°≈ −0.4636
arccot(−1/2)≈ 116.5651°≈ 2.0344

See Also

Recalculate with different values ↑

Arccosine of nearby values

xarccos x
−√3/2150° = 5π/6
−4/5≈ 143.1301°
−√2/2135° = 3π/4
−1/2120° = 2π/3
−0.3≈ 107.4576°
090° = π/2
0.1≈ 84.2608°

FAQ

What is arccos(−1/2)?

arccos(−1/2) = 120° = 2π/3

Why is arccos(−1/2) = 120° and not −60°?

The range of arccos is from 0° to 180°: there are no negative angles in it. A negative x gives an angle greater than 90°: arccos(−1/2) = 180° − arccos(1/2) = 180° − 60° = 120°. The angle −60° does not fit: cos(−α) = cos α, so its cosine is positive.

Why does a calculator show 2.0944 for arccos(−1/2)?

The calculator is in radian mode (RAD) and gives the angle in radians: arccos(−1/2) = 2π/3 ≈ 2.0944 rad. In degrees (DEG) it is 120°.

See Also