arccos12

arccos(12) refers to the inverse cosine of the real number 12. In real-valued trigonometry, the domain of the cosine's inverse is [-1, 1], so arccos(12) is not a real number and is considered undefined in that setting. In complex analysis, however, arccos can be extended to all complex arguments, yielding a complex value.

One common principal-valued formula for complex arguments is arccos z = -i ln(z + sqrt(z^2 - 1)), where sqrt and ln use their principal branches. For z = 12, sqrt(12^2 - 1) = sqrt(143) ≈ 11.9583, and z + sqrt(z^2 - 1) ≈ 23.9583. Taking the natural logarithm gives ln(23.9583) ≈ 3.1763. Therefore, the principal value is arccos(12) ≈ -3.1763 i, a purely imaginary number.

An equivalent expression is arccos z = π/2 − arcsin z, or arccos z = −i acosh z, where acosh is the inverse hyperbolic cosine. For z > 1, acosh z = ln(z + sqrt(z^2 − 1)), so arccos(12) = −i acosh(12) ≈ −3.1763 i. The result reflects the extension of the inverse cosine to complex inputs, with the real part generically absent on real arguments outside the [-1, 1] interval.

In summary, arccos(12) is not real; its principal complex value is approximately −3.1763 i, and it can be expressed as −i ln(12 + sqrt(143)) or −i acosh(12).