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    Sin 105 Degrees

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

    What is sin 105°?

    Answer

    sin 105° =
    (√6 + √2)/4 ≈ 0.9659

    Angle in radians: 105° = 7π/12. The point of the angle is in quadrant II, where sine is positive.

    ≈ — rounded to 4 decimal places.

    sin 105° = sin(180° − 75°) = sin 75° = (√6 + √2)/4

    P(cos 105°, sin 105°) = (−(√6 − √2)/4, (√6 + √2)/4)

    sin 75° = sin(45° + 30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

    xyOP

    P(cos 105°, sin 105°) ≈ (−0.2588, 0.9659)

    Highlighted: the projection on the y-axis — sine, y ≈ 0.9659.

    All four functions of 105°
    FunctionExact valueDecimal
    sin 105°(√6 + √2)/4≈ 0.9659
    cos 105°−(√6 − √2)/4≈ −0.2588
    tan 105°−2 − √3≈ −3.7321
    cot 105°√3 − 2≈ −0.2679

    Reference angle: sin 75° = (√6 + √2)/4

    See Also

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    https://calculat.io/en/number/trigonometry/sin--105Copy
    <a href="https://calculat.io/en/number/trigonometry/sin--105">Sin 105 Degrees: Sine of 105° on the Unit Circle</a>Copy
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    Sin 105 Degrees: Sine of 105° on the Unit Circle. sin 105° = (√6 + √2)/4 ≈ 0.9659. The sine is the y-coordinate of the point at 105° on the unit circle; 105° reduces to 75°.
    https://calculat.io/en/number/trigonometry/sin--105/generated.jpgCopy

    Sine of nearby angles

    Anglesin
    98°≈ 0.9903
    100°≈ 0.9848
    101°≈ 0.9816
    105°(√6 + √2)/4
    106°≈ 0.9613
    108°√(10 + 2√5)/4
    110°≈ 0.9397
    Recalculate with different values ↑

    FAQ

    What is sin 105°?

    sin 105° = (√6 + √2)/4 ≈ 0.9659

    How are sin 15° and cos 75° related?

    They are equal: sin α = cos(90° − α), and 15° + 75° = 90°. So sin 15° = cos 75° = (√6 − √2)/4, cos 15° = sin 75° = (√6 + √2)/4, and tan 15° = cot 75° = 2 − √3.

    Why does a calculator show −0.9705 for sin 105?

    The calculator is in radian mode (RAD): it takes sin 105 as the sine of 105 radians, sin 105 rad ≈ −0.9705. Switch it to degrees (DEG): sin 105° = (√6 + √2)/4 ≈ 0.9659. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also