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    Cos 105 Degrees

    Sin, Cos, Tan, Cot of an Angle
    Function
    Angle (°)

    What is cos 105°?

    Answer

    cos 105° =
    −(√6 − √2)/4 ≈ −0.2588

    Angle in radians: 105° = 7π/12. The point of the angle is in quadrant II, where cosine is negative.

    ≈ — rounded to 4 decimal places.

    cos 105° = cos(180° − 75°) = −cos 75° = −(√6 − √2)/4

    P(cos 105°, sin 105°) = (−(√6 − √2)/4, (√6 + √2)/4)

    cos 75° = cos(45° + 30°) = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4

    xyOP

    P(cos 105°, sin 105°) ≈ (−0.2588, 0.9659)

    Highlighted: the projection on the x-axis — cosine, x ≈ −0.2588.

    All four functions of 105°
    FunctionExact valueDecimal
    sin 105°(√6 + √2)/4≈ 0.9659
    cos 105°−(√6 − √2)/4≈ −0.2588
    tan 105°−2 − √3≈ −3.7321
    cot 105°√3 − 2≈ −0.2679

    Reference angle: cos 75° = (√6 − √2)/4

    See Also

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    https://calculat.io/en/number/trigonometry/cos--105Copy
    <a href="https://calculat.io/en/number/trigonometry/cos--105">Cos 105 Degrees: Cosine of 105° on the Unit Circle</a>Copy
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    Cos 105 Degrees: Cosine of 105° on the Unit Circle. cos 105° = −(√6 − √2)/4 ≈ −0.2588. The cosine is the x-coordinate of the point at 105° on the unit circle; 105° reduces to 75°.
    https://calculat.io/en/number/trigonometry/cos--105/generated.jpgCopy

    Cosine of nearby angles

    Anglecos
    98°≈ −0.1392
    100°≈ −0.1736
    101°≈ −0.1908
    105°−(√6 − √2)/4
    106°≈ −0.2756
    108°−(√5 − 1)/4
    110°≈ −0.3420
    Recalculate with different values ↑

    FAQ

    What is cos 105°?

    cos 105° = −(√6 − √2)/4 ≈ −0.2588

    How are sin 15° and cos 75° related?

    They are equal: sin α = cos(90° − α), and 15° + 75° = 90°. So sin 15° = cos 75° = (√6 − √2)/4, cos 15° = sin 75° = (√6 + √2)/4, and tan 15° = cot 75° = 2 − √3.

    Why does a calculator show −0.2410 for cos 105?

    The calculator is in radian mode (RAD): it takes cos 105 as the cosine of 105 radians, cos 105 rad ≈ −0.2410. Switch it to degrees (DEG): cos 105° = −(√6 − √2)/4 ≈ −0.2588. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also