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    Tan 105 Degrees

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

    What is tan 105°?

    Answer

    tan 105° =
    −2 − √3 ≈ −3.7321

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

    ≈ — rounded to 4 decimal places.

    tan 105° = tan(180° − 75°) = −tan 75° = −2 − √3

    tan 105° = sin 105° / cos 105° = ((√6 + √2)/4) / (−(√6 − √2)/4) = −2 − √3

    tan 75° = tan(45° + 30°) = (1 + √3/3) / (1 − √3/3) = 2 + √3

    xyOP

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

    T(1, tan 105°) ≈ (1, −3.7321) — where line OP meets the line x = 1, on the extension of the ray beyond O (dashed); T is beyond the drawing, the arrow points to it.

    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: tan 75° = 2 + √3

    See Also

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    https://calculat.io/en/number/trigonometry/tan--105Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--105">Tan 105 Degrees: Tangent of 105° on the Unit Circle</a>Copy
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    Tan 105 Degrees: Tangent of 105° on the Unit Circle. tan 105° = −2 − √3 ≈ −3.7321. The tangent is the ratio y/x of the point at 105° on the unit circle; 105° reduces to 75°.
    https://calculat.io/en/number/trigonometry/tan--105/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    98°≈ −7.1154
    100°≈ −5.6713
    101°≈ −5.1446
    105°−2 − √3
    106°≈ −3.4874
    108°−√(5 + 2√5)
    110°≈ −2.7475
    Recalculate with different values ↑

    FAQ

    What is tan 105°?

    tan 105° = −2 − √3 ≈ −3.7321

    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 4.0278 for tan 105?

    The calculator is in radian mode (RAD): it takes tan 105 as the tangent of 105 radians, tan 105 rad ≈ 4.0278. Switch it to degrees (DEG): tan 105° = −2 − √3 ≈ −3.7321. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also