3.1 Trig Functions

Trigonometric Functions

\cos z = \frac{e^{i z} + e^{- i z}}{2}, \sin z = \frac{e^{i z} - e^{- i z}}{2 i} .

We can conclude:

\cos z = 1 - \frac{z^{2}}{2!} + \frac{z^{4}}{4!} - \dots

\sin z = z - \frac{z^{3}}{3!} + \frac{z^{5}}{5!} - \dots

From the definition:

e^{i z} = \cos z + i \sin z

and

\cos^{2} (z) + \sin^{2} (z) = 1.

From the definitions

\sin z = \frac{e^{i z} - e^{- i z}}{2 i}, \cos z = \frac{e^{i z} + e^{- i z}}{2},

we have

(\sin z)^{'} = \cos z, (\cos z)^{'} = - \sin z .

Differentiate using $(e^{i z})^{'} = i e^{i z}, (e^{- i z})^{'} = - i e^{- i z}$ :

(\sin z)^{'} = \frac{i e^{i z} - (- i) e^{- i z}}{2 i} = \frac{e^{i z} + e^{- i z}}{2} = \cos z

(\cos z)^{'} = \frac{i e^{i z} + (- i) e^{- i z}}{2} = \frac{i (e^{i z} - e^{- i z})}{2} = - \frac{e^{i z} - e^{- i z}}{2 i} = - \sin z

Using Euler's formula $e^{i θ} = \cos θ + i \sin θ$ ,

e^{i (α + β)} = e^{i α} e^{i β} = (\cos α + i \sin α) (\cos β + i \sin β) .

Comparing real and imaginary parts gives

\cos (α + β) = \cos α \cos β - \sin α \sin β,

\sin (α + β) = \sin α \cos β + \cos α \sin β .

\tan z = \frac{\sin z}{\cos z} = \frac{\frac{e^{i z} - e^{- i z}}{2 i}}{\frac{e^{i z} + e^{- i z}}{2}} = - i \frac{e^{i z} - e^{- i z}}{e^{i z} + e^{- i z}} .

All other trigonometric functions are rational functions of $e^{i z}$ .