Arccos of −√3/2

Arcsin, Arccos, Arctan, Arccot
Function
x

What is arccos(−√3/2)?

Answer

arccos(−√3/2) =
150° = 5π/6

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

arccos(−√3/2) is the angle α in [0°, 180°] with cos α = −√3/2.

arccos(−√3/2) = 180° − arccos(√3/2) = 180° − 30° = 150°

cos 150° = −√3/2 ✓

xyOP

α = arccos(−√3/2) = 150°

P(cos 150°, sin 150°) = (−√3/2, 1/2)

P lies on the line x = −√3/2 in the upper half of the circle, where sin α ≥ 0; highlighted: cos α = −√3/2.

All four inverse functions of −√3/2
FunctionDegreesRadians
arcsin(−√3/2)−60°−π/3
arccos(−√3/2)150°5π/6
arctan(−√3/2)≈ −40.8934°≈ −0.7137
arccot(−√3/2)≈ 130.8934°≈ 2.2845

See Also

Recalculate with different values ↑

Arccosine of nearby values

xarccos x
−1180° = π
−√3/2150° = 5π/6
−4/5≈ 143.1301°
−√2/2135° = 3π/4
−1/2120° = 2π/3
−0.3≈ 107.4576°
090° = π/2

FAQ

What is arccos(−√3/2)?

arccos(−√3/2) = 150° = 5π/6

Why is arccos(−√3/2) = 150° and not −30°?

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(−√3/2) = 180° − arccos(√3/2) = 180° − 30° = 150°. The angle −30° does not fit: cos(−α) = cos α, so its cosine is positive.

Why does a calculator show 2.6180 for arccos(−√3/2)?

The calculator is in radian mode (RAD) and gives the angle in radians: arccos(−√3/2) = 5π/6 ≈ 2.6180 rad. In degrees (DEG) it is 150°.

See Also