Mémo Trigonométrie

Définition
Identités Remarquables
Cercle Trigonométrique
- Propriétés liées au cercle Trigonométrique

Définition

Une définition de la trigonométrie se base sur les triangles rectangles. Dans ce cas, on a :

\[\sin\alpha = \frac{a}{c} \quad\quad\quad \cos\alpha = \frac{b}{c} \quad\quad\quad \tan\alpha = \frac{a}{b}\]

Pythagore ∶	$a^2+b^2 = c^2$
Al-Kashi ∶	$a = \sqrt{b^2+c^2-2\times b\times c\times \cos\alpha}$
Thalès ∶	$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$

Identités Remarquables

Quel que soit l’angle $\alpha$, on a (d’après le théorème de Pythagore) : $\cos^2⁡\alpha + \sin^2\alpha = 1$

Formules d’addition et de différence des arcs

$\sin⁡(\alpha -\beta )$	$=$	$\sin⁡\alpha \times \cos⁡\beta - \cos⁡\alpha \times \sin⁡\beta$	$\quad \quad \quad$	$\sin⁡(\alpha + \beta )$	$=$	$\sin⁡\alpha \times \cos⁡\beta + \cos⁡\alpha \times \sin⁡\beta\quad$
$\cos(\alpha -\beta )$	$=$	$\cos\alpha \times \cos⁡\beta + \sin\alpha \times \sin⁡\beta$	$\quad \quad \quad$	$\cos(\alpha + \beta )$	$=$	$\cos\alpha \times \cos⁡\beta - \sin\alpha \times \sin⁡\beta\quad$
$\tan(\alpha -\beta )$	$=$	$\left(\tan\alpha - \tan\beta\right)/\left(1 + \tan\alpha \times \tan\beta\right)$	$\quad \quad \quad$	$\tan(\alpha + \beta )$	$=$	$\left(\tan\alpha - \tan\beta\right)/\left(1 + \tan\alpha \times \tan\beta\right)\quad$

Formules de duplication des arcs

$\cos(2\alpha)$	$=$	$\begin{cases} \cos^2⁡\alpha - \sin^2\alpha \\ 2\times \cos^2⁡\alpha - 1 \\ 1 - 2 \times \sin^2\alpha \end{cases}$	$\quad \quad \quad$	$\cos(3\alpha)$	$=$	$4\times \cos^3⁡\alpha-3 \times \cos\alpha$
$\sin(2\alpha)$	$=$	$2\times \sin \alpha\times \cos\alpha$	$\quad \quad \quad$	$\sin(3\alpha)$	$=$	$3 \times \sin \alpha - 4 \times \sin^3⁡\alpha$
$\tan⁡(2\alpha)$	$=$	$\left(2 \times \tan⁡\alpha\right)/\left(1-\tan^2⁡\alpha\right)$	$\quad \quad \quad$	$\tan⁡(3\alpha)$	$=$	$\left(3 \times \tan⁡\alpha - \tan^3⁡\alpha\right)/\left(1 - 3 \times \tan^2⁡\alpha\right)\quad\quad\quad$

Formules d’addition et de différence de deux sinus et de deux cosinus converties en produit

$\cos(\alpha + \beta ) - \cos(\alpha - \beta )$	$=$	$- 2 \times \sin\alpha \times \sin\beta$	$\quad \quad \quad$	$\cos(\alpha + \beta ) + \cos(\alpha - \beta )$	$=$	$2 \times \cos\alpha \times \cos⁡\beta\quad\quad\quad$
$\sin(\alpha + \beta ) - \sin(\alpha - \beta )$	$=$	$2 \times \cos\alpha \times \sin\beta$	$\quad \quad \quad$	$\sin(\alpha + \beta ) + \sin(\alpha - \beta )$	$=$	$2 \times \sin\alpha \times \cos\beta\quad\quad\quad$

Identités trigonométrique

\[\tan\alpha = \frac{\sin\alpha}{\cos\alpha} \quad\quad\quad \cos^2⁡\alpha + \sin^2\alpha = 1 \quad\quad\quad \tan^2⁡\alpha + 1 = \frac{1}{\cos^2\alpha}\]

Relation entre les fonctions trigonométriques

\[\cos\alpha = \sqrt{1-\sin^2\alpha} = \frac{1}{\sqrt{1+\tan^2\alpha}}\\ \sin\alpha = \sqrt{1-\cos^2\alpha} = \frac{\tan\alpha}{\sqrt{1+\tan^2\alpha}}\\ \tan\alpha = \frac{\sqrt{1-\cos^2\alpha}}{\cos\alpha} = \frac{\sin\alpha}{\sqrt{1-\sin^2\alpha}}\]

Cercle Trigonométrique

Radian	$$0$$	$$\frac{\pi}{6}$$	$$\frac{\pi}{4}$$	$$\frac{\pi}{3}$$	$$\frac{\pi}{2}$$	$$\frac{2\pi}{3}$$	$$\frac{3\pi}{4}$$	$$\frac{5\pi}{6}$$	$$\pi$$	$$\frac{7\pi}{6}$$	$$\frac{5\pi}{4}$$	$$\frac{4\pi}{3}$$	$$\frac{3\pi}{2}$$	$$\frac{5\pi}{3}$$	$$\frac{7\pi}{4}$$	$$\frac{11\pi}{6}$$	$$2\pi$$
Degré	$$\ 0\ $$	$$30$$	$$45$$	$$60$$	$$90$$	$$120$$	$$135$$	$$150$$	$$180$$	$$210$$	$$225$$	$$240$$	$$270$$	$$300$$	$$315$$	$$330$$	$$360$$
$$\sin$$	$$0$$	$$\frac{1}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{\sqrt{3}}{2}$$	$$1$$	$$\frac{\sqrt{3}}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{1}{2}$$	$$0$$	$$-\frac{1}{2}$$	$$-\frac{\sqrt{2}}{2}$$	$$-\frac{\sqrt{3}}{2}$$	$$-1$$	$$-\frac{\sqrt{3}}{2}$$	$$-\frac{\sqrt{2}}{2}$$	$$-\frac{1}{2}$$	$$0$$
$$\cos$$	$$1$$	$$\frac{\sqrt{3}}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{1}{2}$$	$$0$$	$$-\frac{1}{2}$$	$$-\frac{\sqrt{2}}{2}$$	$$-\frac{\sqrt{3}}{2}$$	$$-1$$	$$-\frac{\sqrt{3}}{2}$$	$$-\frac{\sqrt{2}}{2}$$	$$-\frac{1}{2}$$	$$0$$	$$\frac{1}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{\sqrt{3}}{2}$$	$$1$$
$$\tan$$	$$0$$	$$\frac{1}{\sqrt{3}}$$	$$1$$	$$\sqrt{3}$$	$$-$$	$$-\sqrt{3}$$	$$-1$$	$$-\frac{1}{\sqrt{3}}$$	$$0$$	$$\frac{1}{\sqrt{3}}$$	$$1$$	$$\sqrt{3}$$	$$-$$	$$-\sqrt{3}$$	$$-1$$	$$-\frac{1}{\sqrt{3}}$$	$$0$$

Propriétés liées au cercle Trigonométrique

Propriété		Sinus				Cosinus				Tangente
Réflexion d’axe $a=0$	$\quad$	$\sin(-\alpha)$	$=$	$-\sin\alpha$	$\quad$	$\cos⁡(-\alpha)$	$=$	$\cos⁡\alpha$	$\quad$	$\tan(-\alpha)$	$=$	$-\tan\alpha\quad\quad$
Réflexion d’axe $a=\pi/4$	$\quad$	$\sin(\pi/2-\alpha)$	$=$	$\cos⁡\alpha$	$\quad$	$\cos⁡(\pi/2-\alpha)$	$=$	$\sin\alpha$	$\quad$	$\tan(\pi/2-\alpha)$	$=$	$\cot⁡\alpha\quad\quad$
Réflexion d’axe $a=\pi/2$	$\quad$	$\sin(\pi-\alpha)$	$=$	$\sin\alpha$	$\quad$	$\cos⁡(\pi-\alpha)$	$=$	$-\cos⁡\alpha$	$\quad$	$\tan(\pi-\alpha)$	$=$	$-\tan\alpha\quad\quad$
Décalage de $\pi/2$	$\quad$	$\sin(\alpha+\pi/2)$	$=$	$\cos⁡\alpha$	$\quad$	$\cos⁡(\alpha+\pi/2)$	$=$	$-\sin\alpha$	$\quad$	$\tan(\alpha+\pi/2)$	$=$	$-\cot⁡\alpha\quad\quad$
Décalage de $\pi$	$\quad$	$\sin(\alpha+\pi)$	$=$	$-\sin\alpha$	$\quad$	$\cos⁡(\alpha+\pi)$	$=$	$-\cos⁡\alpha$	$\quad$	$\tan(\alpha+\pi)$	$=$	$\tan\alpha\quad\quad$
Décalage de $2\pi$	$\quad$	$\sin(\alpha+2\pi)$	$=$	$\sin\alpha$	$\quad$	$\cos⁡(\alpha+2\pi)$	$=$	$\cos⁡\alpha$	$\quad$	$\tan(\alpha+2\pi)$	$=$	$\tan\alpha\quad\quad$

Thibaut Monseigne

Ingénieur Recherche et Développement en Interface Cerveau-Machine.

\(\sin⁡(\alpha -\beta )\)	\(=\)	\(\sin⁡\alpha \times \cos⁡\beta - \cos⁡\alpha \times \sin⁡\beta\)	\(\quad \quad \quad\)	\(\sin⁡(\alpha + \beta )\)	\(=\)	\(\sin⁡\alpha \times \cos⁡\beta + \cos⁡\alpha \times \sin⁡\beta\quad\)
\(\cos(\alpha -\beta )\)	\(=\)	\(\cos\alpha \times \cos⁡\beta + \sin\alpha \times \sin⁡\beta\)	\(\quad \quad \quad\)	\(\cos(\alpha + \beta )\)	\(=\)	\(\cos\alpha \times \cos⁡\beta - \sin\alpha \times \sin⁡\beta\quad\)
\(\tan(\alpha -\beta )\)	\(=\)	\(\left(\tan\alpha - \tan\beta\right)/\left(1 + \tan\alpha \times \tan\beta\right)\)	\(\quad \quad \quad\)	\(\tan(\alpha + \beta )\)	\(=\)	\(\left(\tan\alpha - \tan\beta\right)/\left(1 + \tan\alpha \times \tan\beta\right)\quad\)

\(\cos(2\alpha)\)	\(=\)	\(\begin{cases} \cos^2⁡\alpha - \sin^2\alpha \\ 2\times \cos^2⁡\alpha - 1 \\ 1 - 2 \times \sin^2\alpha \end{cases}\)	\(\quad \quad \quad\)	\(\cos(3\alpha)\)	\(=\)	\(4\times \cos^3⁡\alpha-3 \times \cos\alpha\)
\(\sin(2\alpha)\)	\(=\)	\(2\times \sin \alpha\times \cos\alpha\)	\(\quad \quad \quad\)	\(\sin(3\alpha)\)	\(=\)	\(3 \times \sin \alpha - 4 \times \sin^3⁡\alpha\)
\(\tan⁡(2\alpha)\)	\(=\)	\(\left(2 \times \tan⁡\alpha\right)/\left(1-\tan^2⁡\alpha\right)\)	\(\quad \quad \quad\)	\(\tan⁡(3\alpha)\)	\(=\)	\(\left(3 \times \tan⁡\alpha - \tan^3⁡\alpha\right)/\left(1 - 3 \times \tan^2⁡\alpha\right)\quad\quad\quad\)

\(\cos(\alpha + \beta ) - \cos(\alpha - \beta )\)	\(=\)	\(- 2 \times \sin\alpha \times \sin\beta\)	\(\quad \quad \quad\)	\(\cos(\alpha + \beta ) + \cos(\alpha - \beta )\)	\(=\)	\(2 \times \cos\alpha \times \cos⁡\beta\quad\quad\quad\)
\(\sin(\alpha + \beta ) - \sin(\alpha - \beta )\)	\(=\)	\(2 \times \cos\alpha \times \sin\beta\)	\(\quad \quad \quad\)	\(\sin(\alpha + \beta ) + \sin(\alpha - \beta )\)	\(=\)	\(2 \times \sin\alpha \times \cos\beta\quad\quad\quad\)

Propriété		Sinus				Cosinus				Tangente
Réflexion d’axe \(a=0\)	\(\quad\)	\(\sin(-\alpha)\)	\(=\)	\(-\sin\alpha\)	\(\quad\)	\(\cos⁡(-\alpha)\)	\(=\)	\(\cos⁡\alpha\)	\(\quad\)	\(\tan(-\alpha)\)	\(=\)	\(-\tan\alpha\quad\quad\)
Réflexion d’axe \(a=\pi/4\)	\(\quad\)	\(\sin(\pi/2-\alpha)\)	\(=\)	\(\cos⁡\alpha\)	\(\quad\)	\(\cos⁡(\pi/2-\alpha)\)	\(=\)	\(\sin\alpha\)	\(\quad\)	\(\tan(\pi/2-\alpha)\)	\(=\)	\(\cot⁡\alpha\quad\quad\)
Réflexion d’axe \(a=\pi/2\)	\(\quad\)	\(\sin(\pi-\alpha)\)	\(=\)	\(\sin\alpha\)	\(\quad\)	\(\cos⁡(\pi-\alpha)\)	\(=\)	\(-\cos⁡\alpha\)	\(\quad\)	\(\tan(\pi-\alpha)\)	\(=\)	\(-\tan\alpha\quad\quad\)
Décalage de \(\pi/2\)	\(\quad\)	\(\sin(\alpha+\pi/2)\)	\(=\)	\(\cos⁡\alpha\)	\(\quad\)	\(\cos⁡(\alpha+\pi/2)\)	\(=\)	\(-\sin\alpha\)	\(\quad\)	\(\tan(\alpha+\pi/2)\)	\(=\)	\(-\cot⁡\alpha\quad\quad\)
Décalage de \(\pi\)	\(\quad\)	\(\sin(\alpha+\pi)\)	\(=\)	\(-\sin\alpha\)	\(\quad\)	\(\cos⁡(\alpha+\pi)\)	\(=\)	\(-\cos⁡\alpha\)	\(\quad\)	\(\tan(\alpha+\pi)\)	\(=\)	\(\tan\alpha\quad\quad\)
Décalage de \(2\pi\)	\(\quad\)	\(\sin(\alpha+2\pi)\)	\(=\)	\(\sin\alpha\)	\(\quad\)	\(\cos⁡(\alpha+2\pi)\)	\(=\)	\(\cos⁡\alpha\)	\(\quad\)	\(\tan(\alpha+2\pi)\)	\(=\)	\(\tan\alpha\quad\quad\)

Pythagore ∶	\(a^2+b^2 = c^2\)
Al-Kashi ∶	\(a = \sqrt{b^2+c^2-2\times b\times c\times \cos\alpha}\)
Thalès ∶	\(\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}\)