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    Sin 54 Degrees

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

    What is sin 54°?

    Answer

    sin 54° =
    (√5 + 1)/4 ≈ 0.8090

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

    ≈ — rounded to 4 decimal places.

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

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

    sin 54° = cos 36° = (√5 + 1)/4

    xyOP

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

    Highlighted: the projection on the y-axis — sine, y ≈ 0.8090.

    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/sin--54Copy
    <a href="https://calculat.io/en/number/trigonometry/sin--54">Sin 54 Degrees: Sine of 54° on the Unit Circle - Calculatio</a>Copy
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    Sin 54 Degrees: Sine of 54° on the Unit Circle. sin 54° = (√5 + 1)/4 ≈ 0.8090. The sine is the y-coordinate of the point at 54° on the unit circle.
    https://calculat.io/en/number/trigonometry/sin--54/generated.jpgCopy

    Sine of nearby angles

    Anglesin
    51°≈ 0.7771
    52°≈ 0.7880
    53°≈ 0.7986
    54°(√5 + 1)/4
    55°≈ 0.8192
    56°≈ 0.8290
    57°≈ 0.8387
    Recalculate with different values ↑

    FAQ

    What is sin 54°?

    sin 54° = (√5 + 1)/4 ≈ 0.8090

    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 −0.5588 for sin 54?

    The calculator is in radian mode (RAD): it takes sin 54 as the sine of 54 radians, sin 54 rad ≈ −0.5588. Switch it to degrees (DEG): sin 54° = (√5 + 1)/4 ≈ 0.8090. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also