Trig Functions of Complex Numbers

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:

Angle sum formulae:

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)

[nathanisbored]

everybody likes to talk about *e^ix = cosx + isinx*, but people never talk about *e^x = coshx + sinhx* which I think is equally as cool. you can even compare odd parts and even parts and immediately get that *cosh(ix) = cosx* and *cos(ix) = coshx* which is my favorite way to relate regular and hyperbolic trig functions

[昆仑云路]

Some one asked how to solve sin(x)＝2. This is easy.

[aytunch]

Great video as usual Doc. These kind of simplifications have huge advantages in computer science where we can cache temporary results of similar terms.

[jameswaller]

Thanks so much to make this smooth as butter :)

[mathpuzzles]

Great explanation! Thanks for making it!

[-TheChosenOne]

what an amazing channel I just found!!Keep it up

[StephanieLDiaz]

omgggg thank you so much for doing this!!

[Jkauppa]

try 4d sphere complex solutions: x^2 + y^2 + z^2 = 1 - t^2, where t is the "time" 4th dimension, think specifically the case of |t|>1, where the "length" of the 3d sphere is less than zero

[bjornfeuerbacher]

Having a complex number in the denominator isn't nice, one should multiply numerator and denominator in the expressions for tan, cot, sec and csc with the complex conjugate. But I suspect that makes the expressions much more clumsier. :/

[frentz]

You never defined what "sin (a + bi)" supposedly *means*. What is the point of developing formulas for a thing that has no meaning, that is undefined? [ie for real numbers theta, we have a meaning of "sin (theta)" as coming from a point (x, y) on the unit circle. but what's "sin (z)" for a complex value z ?]