Computing sin(i)

Complex Analysis · Axiom Academy

Using the exponential definition of sine to evaluate the sine of an imaginary number. Compute , the sine of the imaginary unit, and write the answer in exact form. For real angles, never leaves the band . But is purely imaginary with magnitude , so it sits on the imaginary axis — outside the unit band that bounds real sine. Nice work — you computed the sine of an imaginary number straight from the exponential definition. Sine has an exponential definition: works for every complex z , not just real angles. The imaginary unit collapses the exponents: , which turns e^ iz and e^ -iz into the real numbers e^ -1 and e^ 1 . What's left is hyperbolic sine: . Result: — purely imaginary, with magnitude greater than 1 . Real sine is trapped in [-1, 1] , but complex sine is unbounded: feeding it an imaginary input pushes the output off the unit band entirely.

This is the written version of the interactive lesson above. See the full Complex Analysis course.