arccos15

arccos15 refers to the inverse cosine function applied to the value 15. Because the real‐valued inverse cosine is defined only for inputs in the interval [–1, 1], the expression arccos15 is not defined over the real numbers. Instead, mathematicians use the analytic continuation of the arccos function to the complex plane. In this extended definition, arccos z is a multi‑valued complex function that can be expressed as

arccos z = –i ln(z + √(z²–1)),

where the principal value of the complex logarithm and the square root are taken. For z = 15 this yields

arccos 15 = –i ln(15 + √(225 – 1))

= –i ln(15 + √224)

≈ –i ln(15 + 14.9666)

≈ –i (2.682).

Thus the principal value of arccos15 is purely imaginary, approximately –2.682 i. The function has infinitely many other values differing by integer multiples of 2π due to the periodicity of the complex exponential.

The complex arccos is closely related to the inverse hyperbolic cosine, arccosh. In fact, for real x > 1 one has

arccos x = i arccosh x,

which follows from substituting the definition of arccosh into the complex logarithm formula. Consequently, the magnitude of the imaginary part of arccos15 is equal to arccosh15, whose numeric value is about 2.682.

In practical applications, such as solving equations involving trigonometric identities with arguments outside [–1, 1], the complex arccos may be employed. It is also encountered in analytic continuation of Fourier transforms and in certain branches of theoretical physics where complex angles arise. Nevertheless, for most elementary contexts only the real‑valued arccos function, defined on [–1, 1], is relevant, and arccos15 is typically noted as undefined in that domain.