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    Tan 54 Degrees

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

    What is tan 54°?

    Answer

    tan 54° =
    √(25 + 10√5)/5 ≈ 1.3764

    Angle in radians: 54° = 3π/10. The point of the angle is in quadrant I, where tangent is positive.

    ≈ — rounded to 4 decimal places.

    tan 54° = sin 54° / cos 54° = ((√5 + 1)/4) / (√(10 − 2√5)/4) = √(25 + 10√5)/5

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

    tan 54° = cot 36° = cos 36° / sin 36° = √(25 + 10√5)/5

    xyOP

    P(cos 54°, sin 54°) ≈ (0.5878, 0.8090)

    T(1, tan 54°) ≈ (1, 1.3764) — where line OP meets the line x = 1.

    All four functions of 54°
    FunctionExact valueDecimal
    sin 54°(√5 + 1)/4≈ 0.8090
    cos 54°√(10 − 2√5)/4≈ 0.5878
    tan 54°√(25 + 10√5)/5≈ 1.3764
    cot 54°√(5 − 2√5)≈ 0.7265

    See Also

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    https://calculat.io/en/number/trigonometry/tan--54Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--54">Tan 54 Degrees: Tangent of 54° on the Unit Circle</a>Copy
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    Tan 54 Degrees: Tangent of 54° on the Unit Circle. tan 54° = √(25 + 10√5)/5 ≈ 1.3764. The tangent is the ratio y/x of the point at 54° on the unit circle.
    https://calculat.io/en/number/trigonometry/tan--54/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    51°≈ 1.2349
    52°≈ 1.2799
    53°≈ 1.3270
    54°√(25 + 10√5)/5
    55°≈ 1.4281
    56°≈ 1.4826
    57°≈ 1.5399
    Recalculate with different values ↑

    FAQ

    What is tan 54°?

    tan 54° = √(25 + 10√5)/5 ≈ 1.3764

    How are sin 36° and cos 54° related?

    They are equal: sin α = cos(90° − α), and 36° + 54° = 90°. So sin 36° = cos 54° = √(10 − 2√5)/4, cos 36° = sin 54° = (√5 + 1)/4, and tan 36° = cot 54° = √(5 − 2√5).

    Why does a calculator show 0.6738 for tan 54?

    The calculator is in radian mode (RAD): it takes tan 54 as the tangent of 54 radians, tan 54 rad ≈ 0.6738. Switch it to degrees (DEG): tan 54° = √(25 + 10√5)/5 ≈ 1.3764. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also