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    Tan 108 Degrees

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

    What is tan 108°?

    Answer

    tan 108° =
    −√(5 + 2√5) ≈ −3.0777

    Angle in radians: 108° = 3π/5. The point of the angle is in quadrant II, where tangent is negative.

    ≈ — rounded to 4 decimal places.

    tan 108° = tan(180° − 72°) = −tan 72° = −√(5 + 2√5)

    tan 108° = sin 108° / cos 108° = (√(10 + 2√5)/4) / (−(√5 − 1)/4) = −√(5 + 2√5)

    Regular pentagon (golden ratio): cos 36° = (√5 + 1)/4.

    tan 72° = cot 18° = cos 18° / sin 18° = √(5 + 2√5)

    xyOP

    P(cos 108°, sin 108°) ≈ (−0.3090, 0.9511)

    T(1, tan 108°) ≈ (1, −3.0777) — 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 108°
    FunctionExact valueDecimal
    sin 108°√(10 + 2√5)/4≈ 0.9511
    cos 108°−(√5 − 1)/4≈ −0.3090
    tan 108°−√(5 + 2√5)≈ −3.0777
    cot 108°−√(25 − 10√5)/5≈ −0.3249

    Reference angle: tan 72° = √(5 + 2√5)

    See Also

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    https://calculat.io/en/number/trigonometry/tan--108Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--108">Tan 108 Degrees: Tangent of 108° on the Unit Circle</a>Copy
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    Tan 108 Degrees: Tangent of 108° on the Unit Circle. tan 108° = −√(5 + 2√5) ≈ −3.0777. The tangent is the ratio y/x of the point at 108° on the unit circle; 108° reduces to 72°.
    https://calculat.io/en/number/trigonometry/tan--108/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    101°≈ −5.1446
    105°−2 − √3
    106°≈ −3.4874
    108°−√(5 + 2√5)
    110°≈ −2.7475
    120°−√3
    125°≈ −1.4281
    Recalculate with different values ↑

    FAQ

    What is tan 108°?

    tan 108° = −√(5 + 2√5) ≈ −3.0777

    How are sin 18° and cos 72° related?

    They are equal: sin α = cos(90° − α), and 18° + 72° = 90°. So sin 18° = cos 72° = (√5 − 1)/4, cos 18° = sin 72° = √(10 + 2√5)/4, and tan 18° = cot 72° = √(25 − 10√5)/5.

    Why does a calculator show 2.4682 for tan 108?

    The calculator is in radian mode (RAD): it takes tan 108 as the tangent of 108 radians, tan 108 rad ≈ 2.4682. Switch it to degrees (DEG): tan 108° = −√(5 + 2√5) ≈ −3.0777. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also