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    Tan 72 Degrees

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

    What is tan 72°?

    Answer

    tan 72° =
    √(5 + 2√5) ≈ 3.0777

    Angle in radians: 72° = 2π/5. The point of the angle is in quadrant I, where tangent is positive.

    ≈ — rounded to 4 decimal places.

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

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

    tan 72° = cot 18° = cos 18° / sin 18° = √(5 + 2√5)

    xyOP

    P(cos 72°, sin 72°) ≈ (0.3090, 0.9511)

    T(1, tan 72°) ≈ (1, 3.0777) — where line OP meets the line x = 1; T is beyond the drawing, the arrow points to it.

    All four functions of 72°
    FunctionExact valueDecimal
    sin 72°√(10 + 2√5)/4≈ 0.9511
    cos 72°(√5 − 1)/4≈ 0.3090
    tan 72°√(5 + 2√5)≈ 3.0777
    cot 72°√(25 − 10√5)/5≈ 0.3249

    See Also

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    https://calculat.io/en/number/trigonometry/tan--72Copy
    <a href="https://calculat.io/en/number/trigonometry/tan--72">Tan 72 Degrees: Tangent of 72° on the Unit Circle</a>Copy
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    Tan 72 Degrees: Tangent of 72° on the Unit Circle. tan 72° = √(5 + 2√5) ≈ 3.0777. The tangent is the ratio y/x of the point at 72° on the unit circle.
    https://calculat.io/en/number/trigonometry/tan--72/generated.jpgCopy

    Tangent of nearby angles

    Angletan
    69°≈ 2.6051
    70°≈ 2.7475
    71°≈ 2.9042
    72°√(5 + 2√5)
    73°≈ 3.2709
    74°≈ 3.4874
    75°2 + √3
    Recalculate with different values ↑

    FAQ

    What is tan 72°?

    tan 72° = √(5 + 2√5) ≈ 3.0777

    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 −0.2624 for tan 72?

    The calculator is in radian mode (RAD): it takes tan 72 as the tangent of 72 radians, tan 72 rad ≈ −0.2624. Switch it to degrees (DEG): tan 72° = √(5 + 2√5) ≈ 3.0777. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

    See Also