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Showing content from https://dlmf.nist.gov/4.43 below:

§4.43 Cubic Equations ‣ Applications ‣ Chapter 4 Elementary Functions

§4.43 Cubic Equations

Let p ( ≠ 0 ) and q be real constants and

4.43.1 A = ( − 4 3 ⁢ p ) 1 / 2 , B = ( 4 3 ⁢ p ) 1 / 2 .

The roots of

4.43.2 z3 + p ⁢ z + q = 0

are:

  1. (a)

    A ⁢ sin ⁡ a , A ⁢ sin ⁡ ( a + 2 3 ⁢ π ) , and A ⁢ sin ⁡ ( a + 4 3 ⁢ π ) , with sin ⁡ ( 3 ⁢ a ) = 4 ⁢ q / A3 , when 4 ⁢ p3 + 27 ⁢ q2 ≤ 0 .

  2. (b)

    A ⁢ cosh ⁡ a , A ⁢ cosh ⁡ ( a + 2 3 ⁢ π ⁢ i ) , and A ⁢ cosh ⁡ ( a + 4 3 ⁢ π ⁢ i ) , with cosh ⁡ ( 3 ⁢ a ) = − 4 ⁢ q / A3 , when p < 0 , q < 0 , and 4 ⁢ p3 + 27 ⁢ q2 > 0 .

  3. (c)

    B ⁢ sinh ⁡ a , B ⁢ sinh ⁡ ( a + 2 3 ⁢ π ⁢ i ) , and B ⁢ sinh ⁡ ( a + 4 3 ⁢ π ⁢ i ) , with sinh ⁡ ( 3 ⁢ a ) = − 4 ⁢ q / B3 , when p > 0 .

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

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