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    Sin 216 Degrees

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

    What is sin 216°?

    Answer

    sin 216° =
    −√(10 − 2√5)/4 ≈ −0.5878

    Angle in radians: 216° = 6π/5. The point of the angle is in quadrant III, where sine is negative.

    ≈ — rounded to 4 decimal places.

    sin 216° = sin(180° + 36°) = −sin 36° = −√(10 − 2√5)/4

    P(cos 216°, sin 216°) = (−(√5 + 1)/4, −√(10 − 2√5)/4)

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

    sin 36° = √(1 − cos² 36°) = √(10 − 2√5)/4

    xyOP

    P(cos 216°, sin 216°) ≈ (−0.8090, −0.5878)

    Highlighted: the projection on the y-axis — sine, y ≈ −0.5878.

    All four functions of 216°
    FunctionExact valueDecimal
    sin 216°−√(10 − 2√5)/4≈ −0.5878
    cos 216°−(√5 + 1)/4≈ −0.8090
    tan 216°√(5 − 2√5)≈ 0.7265
    cot 216°√(25 + 10√5)/5≈ 1.3764

    Reference angle: sin 36° = √(10 − 2√5)/4

    See Also

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    https://calculat.io/en/number/trigonometry/sin--216Copy
    <a href="https://calculat.io/en/number/trigonometry/sin--216">Sin 216 Degrees: Sine of 216° on the Unit Circle</a>Copy
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    Sin 216 Degrees: Sine of 216° on the Unit Circle. sin 216° = −√(10 − 2√5)/4 ≈ −0.5878. The sine is the y-coordinate of the point at 216° on the unit circle; 216° reduces to 36°.
    https://calculat.io/en/number/trigonometry/sin--216/generated.jpgCopy

    Sine of nearby angles

    Anglesin
    210°−1/2
    211°≈ −0.5150
    215°≈ −0.5736
    216°−√(10 − 2√5)/4
    225°−√2/2
    227°≈ −0.7314
    234°−(√5 + 1)/4
    Recalculate with different values ↑

    FAQ

    What is sin 216°?

    sin 216° = −√(10 − 2√5)/4 ≈ −0.5878

    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 0.6961 for sin 216?

    The calculator is in radian mode (RAD): it takes sin 216 as the sine of 216 radians, sin 216 rad ≈ 0.6961. Switch it to degrees (DEG): sin 216° = −√(10 − 2√5)/4 ≈ −0.5878. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also