the complex trigonometric functions

We know cos and sin as ratios in a right triangle, or as the coordinates of a point going around the unit circle. Neither picture obviously tells you what cos(2 + 3 i) should be. The clean way to extend them to complex inputs is to notice that, on the real line, they were secretly built out of the exponential all along.

Euler's formula gives e^(i t) = cos t + i sin t and e^(-i t) = cos t - i sin t. Adding and subtracting solves for the trig functions: cos t = (e^(i t) + e^(-i t)) / 2 and sin t = (e^(i t) - e^(-i t)) / (2 i). We simply use these same formulas with a complex z in place of t: cos z = (e^(i z) + e^(-i z)) / 2 and sin z = (e^(i z) - e^(-i z)) / (2 i). Because e^z is entire, so are cos z and sin z, and all the familiar identities survive unchanged: sin^2 z + cos^2 z = 1, the derivative of sin z is cos z, the derivative of cos z is -sin z, the angle-addition formulas, and 2 pi periodicity along the real direction.

Splitting z = x + i y and expanding gives the working formulas sin z = sin x cosh y + i cos x sinh y and cos z = cos x cosh y - i sin x sinh y, where cosh and sinh are the hyperbolic functions. These show the new behaviour at a glance: the cosh y and sinh y factors grow without bound as y moves away from the real axis, which is why complex sine and cosine are unbounded even though their real versions stay between -1 and 1.