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    Tan 288 Degrees

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

    What is tan 288°?

    Answer

    tan 288° =
    −√(5 + 2√5) ≈ −3.0777

    Angle in radians: 288° = 8π/5. The point of the angle is in quadrant IV, where tangent is negative.

    ≈ — rounded to 4 decimal places.

    tan 288° = tan(360° − 72°) = −tan 72° = −√(5 + 2√5)

    tan 288° = sin 288° / cos 288° = (−√(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 288°, sin 288°) ≈ (0.3090, −0.9511)

    T(1, tan 288°) ≈ (1, −3.0777) — where line OP meets the line x = 1; T is beyond the drawing, the arrow points to it.

    All four functions of 288°
    FunctionExact valueDecimal
    sin 288°−√(10 + 2√5)/4≈ −0.9511
    cos 288°(√5 − 1)/4≈ 0.3090
    tan 288°−√(5 + 2√5)≈ −3.0777
    cot 288°−√(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--288Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--288">Tan 288 Degrees: Tangent of 288° on the Unit Circle</a>Copy
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    Tan 288 Degrees: Tangent of 288° on the Unit Circle. tan 288° = −√(5 + 2√5) ≈ −3.0777. The tangent is the ratio y/x of the point at 288° on the unit circle; 288° reduces to 72°.
    https://calculat.io/en/number/trigonometry/tan--288/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    255°2 + √3
    270°undefined
    285°−2 − √3
    288°−√(5 + 2√5)
    300°−√3
    306°−√(25 + 10√5)/5
    315°−1
    Recalculate with different values ↑

    FAQ

    What is tan 288°?

    tan 288° = −√(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 −1.6522 for tan 288?

    The calculator is in radian mode (RAD): it takes tan 288 as the tangent of 288 radians, tan 288 rad ≈ −1.6522. Switch it to degrees (DEG): tan 288° = −√(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