We derive nice expressions for trigonometric functions of complex numbers, in terms of trigonometric and hyperbolic functions of real numbers (we do this for cos, sin, tan, sec, cosec, and cot).

Along the way, we derive the exponential form of sin and cos from Euler's formula, as well as the identities cos(ix) = cosh(x) and sin(ix) = i sinh(x).

Euler's formula:
https://en.wikipedia.org/wiki/Euler%2...

Angle sum formulae:
https://en.wikipedia.org/wiki/List_of...

00:00 Intro
00:12 Exponential form of sin and cos
01:50 cos(a+bi)
02:37 cos(ix)
03:23 sin(ix)
04:52 sin(a+bi)
05:47 tan(a+bi)
07:58 sec(a+bi)
10:10 cosec(a+bi)
11:36 cot(a+bi)