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    Arccos of −√2/2

    Arcsin, Arccos, Arctan, Arccot
    Function
    x

    What is arccos(−√2/2)?

    Answer

    arccos(−√2/2) =
    135° = 3π/4

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

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

    arccos(−√2/2) = 180° − arccos(√2/2) = 180° − 45° = 135°

    cos 135° = −√2/2 ✓

    xyOP

    α = arccos(−√2/2) = 135°

    P(cos 135°, sin 135°) = (−√2/2, √2/2)

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

    All four inverse functions of −√2/2
    FunctionDegreesRadians
    arcsin(−√2/2)−45°−π/4
    arccos(−√2/2)135°3π/4
    arctan(−√2/2)≈ −35.2644°≈ −0.6155
    arccot(−√2/2)≈ 125.2644°≈ 2.1863

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    https://calculat.io/en/number/inverse-trigonometry/arccos---sqrt2-2Copy
    <a href="https://calculat.io/en/number/inverse-trigonometry/arccos---sqrt2-2">Arccos of −√2/2 (Inverse Cosine) in Degrees and Radians</a>Copy
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    Arccos of −√2/2 (Inverse Cosine) in Degrees and Radians. arccos(−√2/2) = 135° = 3π/4: the angle in [0°, 180°] whose cosine is −√2/2. With the check and the unit circle.
    https://calculat.io/en/number/inverse-trigonometry/arccos---sqrt2-2/generated.jpgCopy

    FAQ

    What is arccos(−√2/2)?

    arccos(−√2/2) = 135° = 3π/4

    Why is arccos(−√2/2) = 135° and not −45°?

    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(−√2/2) = 180° − arccos(√2/2) = 180° − 45° = 135°. The angle −45° does not fit: cos(−α) = cos α, so its cosine is positive.

    Why does a calculator show 2.3562 for arccos(−√2/2)?

    The calculator is in radian mode (RAD) and gives the angle in radians: arccos(−√2/2) = 3π/4 ≈ 2.3562 rad. In degrees (DEG) it is 135°.