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Definition: Trigonometric Functions
$z \in \C.$
$$\cos z \triangleq \frac{ e^{iz} + e^{-iz}}{ 2 } $$
$$\sin z \triangleq \frac{ e^{iz} - e^{-iz}}{ 2 } $$
Remarks
$$e^{iz} = 1 + iz - \frac{ z^2 }{ 2! } - \frac{ iz^3 }{ 3! } + \frac{ z^4 }{ 4! } + … $$ $$e^{-iz} = 1 - iz - \frac{ z^2 }{ 2! } + \frac{ iz^3 }{ 3! } + \frac{ z^4 }{ 4! } - … $$ $$\cos z = 1 - \frac{ z^2 }{ 2! } + \frac{ z^4 }{ 4! } - … = \sum_{ n=0 }^{ \infty } \frac{(-1)^n z^{2n}}{(2n)! } $$ $$\sin z = 1 - \frac{ z^3 }{ 3! } + \frac{ z^5 }{ 5! } - … = \sum_{ n=0 }^{ \infty } \frac{(-1)^n z^{2n+1}}{(2n+1)! } $$
Theorem
: Trigonometric Identities
Let $x, y$ be real numbers.
- $\sin ^2 x + \cos ^2 x = 1 $
$\sin x \in [-1, 1], \cos x \in [-1, 1], \forall x \in \R $ - $\sin' x = \cos ' x. $
$\cos' x = - \sin x$. - $\sin (-x) = - \sin x$ and $\cos (-x) = \cos x $
- $\cos (x + y) = \cos x \cos y - \sin x \sin y$
$\sin (x + y) = \sin x \cos y + \cos x \sin y $ - $\sin 0 = 0, \cos 0 = 1$
- $e^{ix} = \cos x + i \sin x $
$e^{-ix} = \cos x - i \sin x $
$\cos x = \Re{ e^{ix}}$
$\sin x = \Im{ e^{ix}}$
Lemma
$\exists x > 0, \sin x = 0$.
Definition: \Pi
$\pi \triangleq \infi{ \set{ x \in (0, \infty): \sin x = 0 }} $
Theorem
: Periodicity of Trigonometric Functions
$x \in \R $
- $\cos (x + \pi) = - \cos x $
$\sin (x + \pi) = - \sin x $
$\sin x = 0 \iff \frac{ x }{ \pi } \in \Z$
$\cos x = 0 \iff \frac{ x }{ \pi } - \frac{ 1 }{ 2 } \in \Z $