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    Tan 165 Degrees

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

    What is tan 165°?

    Answer

    tan 165° =
    √3 − 2 ≈ −0.2679

    Angle in radians: 165° = 11π/12. The point of the angle is in quadrant II, where tangent is negative.

    ≈ — rounded to 4 decimal places.

    tan 165° = tan(180° − 15°) = −tan 15° = √3 − 2

    tan 165° = sin 165° / cos 165° = ((√6 − √2)/4) / (−(√6 + √2)/4) = √3 − 2

    tan 15° = tan(45° − 30°) = (1 − √3/3) / (1 + √3/3) = 2 − √3

    xyOP

    P(cos 165°, sin 165°) ≈ (−0.9659, 0.2588)

    T(1, tan 165°) ≈ (1, −0.2679) — where line OP meets the line x = 1, on the extension of the ray beyond O (dashed).

    All four functions of 165°
    FunctionExact valueDecimal
    sin 165°(√6 − √2)/4≈ 0.2588
    cos 165°−(√6 + √2)/4≈ −0.9659
    tan 165°√3 − 2≈ −0.2679
    cot 165°−2 − √3≈ −3.7321

    Reference angle: tan 15° = 2 − √3

    See Also

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    https://calculat.io/en/number/trigonometry/tan--165Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--165">Tan 165 Degrees: Tangent of 165° on the Unit Circle</a>Copy
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    Tan 165 Degrees: Tangent of 165° on the Unit Circle. tan 165° = √3 − 2 ≈ −0.2679. The tangent is the ratio y/x of the point at 165° on the unit circle; 165° reduces to 15°.
    https://calculat.io/en/number/trigonometry/tan--165/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    160°≈ −0.3640
    162°−√(25 − 10√5)/5
    164°≈ −0.2867
    165°√3 − 2
    167°≈ −0.2309
    175°≈ −0.0875
    179°≈ −0.0175
    Recalculate with different values ↑

    FAQ

    What is tan 165°?

    tan 165° = √3 − 2 ≈ −0.2679

    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 −15.0414 for tan 165?

    The calculator is in radian mode (RAD): it takes tan 165 as the tangent of 165 radians, tan 165 rad ≈ −15.0414. Switch it to degrees (DEG): tan 165° = √3 − 2 ≈ −0.2679. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also