From 7af7ed82f42cd73e79d76cfbf0cb841b191046be Mon Sep 17 00:00:00 2001
From: saundersp <pierre.saundersgb@gmail.com>
Date: Tue, 3 Dec 2024 18:15:11 +0100
Subject: [PATCH] contents/trigonometry.tex : Added Euler's formula

---
 contents/trigonometry.tex | 60 +++++++++++++++++++++++++++++++++++++--
 1 file changed, 57 insertions(+), 3 deletions(-)

diff --git a/contents/trigonometry.tex b/contents/trigonometry.tex
index f7989f9..023e6f5 100644
--- a/contents/trigonometry.tex
+++ b/contents/trigonometry.tex
@@ -6,9 +6,15 @@
 
 Le cercle unitaire est un cercle de centre $(0,0)$ et de rayon 1.
 
+$\forall x \in \R, \cos^2 x + \sin^2 x = 1$
+
 \subsection{cos}
 %TODO Complete subsection
 
+Formule d'Euler
+
+$\forall \theta \in \R, cos(\theta) = \frac{e^{i\theta} + e^{-i \theta}}{2}$
+
 $\cos 0 = 1$
 
 $\cos \frac{\pi}{2} = 0$
@@ -17,12 +23,18 @@ $\cos \pi = -1$
 
 $\cos(-\frac{\pi}{2}) = 0$
 
+$\cos(\frac{\pi}{2} + t) = -\sin(t)$
+
 $\cos(\pi + t) = -\cos(t)$
 
+$\cos(\pi - t) = -\cos(t)$
+
 $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$
 
 $\cos\frac{\pi}{3} = \frac{1}{2}$
 
+$\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$
+
 $\forall (a,b) \in \R$
 
 $\cos(a + b) = \cos a \cos b + \sin a \sin b$
@@ -31,20 +43,46 @@ $\cos(a - b) = \cos a \cos b - \sin a \sin b$
 
 $\cos a + \cos b = 2 \cos(\frac{a + b}{2}) \cos(\frac{a - b}{2} )$
 
+$\cos a - \cos b = -2 \sin(\frac{a + b}{2}) \sin(\frac{a - b}{2} )$
+
+$\cos a \cos b = \frac{\cos(a + b) + \cos(a - b)}{2}$
+
+$\forall t \in \R, \cos(\frac{\pi}{2} - t) = \sin(t)$
+
+$\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
+
+$\frac{d}{dx} \cos x = -\sin x$
+
+$\forall x \in [-1, 1], \cos(\arcsin(x)) = \sqrt{1-x^{2}}$
+
+$\forall x \in [-1, 1], \cos(\arccos(x)) = x$
+
+$\forall x\in\R, \cos^2 x = \frac{1 + \cos(2x)}{2}$
+
 \subsection{sin}
 %TODO Complete subsection
 
+Formule d'Euler
+
+$\forall \theta \in \R, sin(\theta) = \frac{e^{i\theta} - e^{-i \theta}}{2i}$
+
 $\sin 0 = 0$
 
-$\sin(\pi - t) = \sin(t)$
+$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$
 
-$\sin(\frac{\pi}{2} - t) = \cos(t)$
+$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$
 
 $\sin \frac{\pi}{6} = \frac{1}{2}$
 
 $\sin \frac{\pi}{2} = 1$
 
-%$\sin(\frac{\pi}{2} + t) = -\cos(t)$
+$\sin(\frac{\pi}{2} + t) = -\cos(t)$
+
+$\sin(\pi - t) = \sin(t)$
+
+$\sin(\pi + t) = -\sin(t)$
+
+$\sin(\frac{\pi}{2} - t) = \cos(t)$
 
 $\forall (a,b) \in \R$
 
@@ -54,13 +92,29 @@ $\sin(a - b) = \sin a \cos b - \sin b \cos a$
 
 $\sin a - \sin b = 2 \cos (\frac{a+b}{2}) \sin (\frac{a-b}{2})$
 
+$\sin a + \sin b = 2 \sin (\frac{a+b}{2}) \cos (\frac{a-b}{2})$
+
 $\sin a\sin b = \frac{\cos(a - b) - \cos(a + b)}{2}$
 
+$\sin a \cos b = \frac{\sin(a - b) - \sin(a + b)}{2}$
+
+$\lim\limits_{t \to 0} \frac{\sin t}{t} = 1$
+
+$\frac{d}{dx} \sin x = \cos x$
+
+$\forall x \in [-1, 1], \sin(\arcsin(x)) = x$
+
+$\forall x \in [-1, 1], \sin(\arccos(x)) = \sqrt{1-x^{2}}$
+
+$\forall x\in\R, \sin^2 x = \frac{1 - \cos(2x)}{2}$
+
 \subsection{tan}
 %TODO Complete subsection
 
 $\tan 0 = 0$
 
+$\tan \frac{\pi}{3} = \sqrt{3}$
+
 $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$
 
 $\tan \frac{\pi}{4} = 1$