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    Sin 108 Degrees

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

    What is sin 108°?

    Answer

    sin 108° =
    √(10 + 2√5)/4 ≈ 0.9511

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

    ≈ — rounded to 4 decimal places.

    sin 108° = sin(180° − 72°) = sin 72° = √(10 + 2√5)/4

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

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

    sin 72° = cos 18° = √(1 − sin² 18°) = √(10 + 2√5)/4

    xyOP

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

    Highlighted: the projection on the y-axis — sine, y ≈ 0.9511.

    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: sin 72° = √(10 + 2√5)/4

    See Also

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    https://calculat.io/en/number/trigonometry/sin--108Copy
    <a href="https://calculat.io/en/number/trigonometry/sin--108">Sin 108 Degrees: Sine of 108° on the Unit Circle</a>Copy
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    Sin 108 Degrees: Sine of 108° on the Unit Circle. sin 108° = √(10 + 2√5)/4 ≈ 0.9511. The sine is the y-coordinate of the point at 108° on the unit circle; 108° reduces to 72°.
    https://calculat.io/en/number/trigonometry/sin--108/generated.jpgCopy

    Sine of nearby angles

    Anglesin
    101°≈ 0.9816
    105°(√6 + √2)/4
    106°≈ 0.9613
    108°√(10 + 2√5)/4
    110°≈ 0.9397
    120°√3/2
    125°≈ 0.8192
    Recalculate with different values ↑

    FAQ

    What is sin 108°?

    sin 108° = √(10 + 2√5)/4 ≈ 0.9511

    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 0.9268 for sin 108?

    The calculator is in radian mode (RAD): it takes sin 108 as the sine of 108 radians, sin 108 rad ≈ 0.9268. Switch it to degrees (DEG): sin 108° = √(10 + 2√5)/4 ≈ 0.9511. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also