Trigonométrie

Cercle trigo

\[ 0 \] \[ \frac{\pi}{6} \] \[ \frac{\pi}{4} \] \[ \frac{\pi}{3} \] \[ \frac{\pi}{2} \]
\[ \sin \] \[ 0 \] \[ \frac{1}{2} \] \[ \frac{1}{\sqrt{2}} \] \[ \frac{\sqrt{3}}{2} \] \[ 1 \]
\[ \tan \] \[ 0 \] \[ \frac{1}{\sqrt{3}} \] \[ 1 \] \[ \frac{\sqrt{3}}{1} \]
\( \cos \) c'est \( \sin \) à l'envers
\( \cot \) c'est \( \tan \) à l'envers
\( \sec \) c'est l'inverse de \( \cos \)
\( \csc \) c'est l'inverse de \( \sin \)
\[ 0° \] \[ 30° \] \[ 45° \] \[ 60° \] \[ 90° \]
\[ \sin \] \[ 0 \] \[ 0.5 \] \[ 0.707 \] \[ 0.866 \] \[ 1 \]
\[ \tan \] \[ 0 \] \[ 0.577 \] \[ 1 \] \[ 1.732 \]
valeurs à 3 décimales approximatives

Identités trigonométriques

cos-adj-hyp sin-opp-hyp tang-opp-ad
\[ \cos \theta = \frac{\text{adj}}{\text{hyp}} \quad\quad \sin \theta = \frac{\text{opp}}{\text{hyp}} \quad\quad \tan \theta = \frac{\text{opp}}{\text{adj}} \]

Identités de quotients

\[ \orange{\frac{sin}{cos}} = \frac{\green{tan}}{1} = \frac{1}{\green{cot}} = \blue{\frac{sec}{csc}} \]

Sommes

si, si, coco, si ! cococo mousse ici tatata amou tata
\[ \begin{aligned} & \sin(\alpha \green\pm \beta) ~~=~~ sin(\alpha) ~ cos(\beta) ~\green\pm~ cos(\alpha) ~ sin(\beta) \\[1em] & \cos(\alpha \green\pm \beta) ~~=~~ cos(\alpha)cos(\beta) ~~\red\mp~~ sin(\alpha)sin(\beta) \\[3em] & \tan(\alpha \green± \beta) ~~=~~ \frac{ \tan(\alpha) \green\pm \tan(\beta) }{ 1 \red\mp \tan(\alpha) \tan(\beta) } \\ \end{aligned} \]
\[ \Rightarrow\qquad \begin{aligned} & \sin(2 \alpha) ~~=~~ 2 \, sin(\alpha) \, \cos(\alpha) \\[1em] & \cos(2 \alpha) ~~=~~ \cos^2(\alpha) - \sin^2(\alpha) \\[1em] & \tan(2 \alpha) ~~=~~ \frac{2 \tan(\alpha)}{1 - \tan^2(\alpha)} \end{aligned} \]

Pythagore

\[ \begin{aligned} & \sin^2 + \cos^2 = 1 \\[1.5em] & \sec^2 - \tan^2 = 1 \\[1.5em] & \csc^2 - \cot^2 = 1 \end{aligned} \]

Rotation 2D

Roter  \( x \)  et  \( y \)  de  \( \delta \)

Approche naïve

\[ \theta = \text{arctan2} (y, x) \quad;\quad r = \sqrt{x^2 + y^2} \] \[ x' = \cos(\theta + \delta) ~ r \quad;\quad y' = \sin(\theta + \delta) ~ r \]

Simplification

Base : \[ \begin{cases} x' = \cos(\theta + \delta) ~ r \\[0.5em] y' = \sin(\theta + \delta) ~ r \end{cases} \] identités des sommes : \[ \begin{cases} x' = r \left( \cos(\theta) \, \cos(\delta) ~~-~~ \sin(\theta) \, \sin(\delta) \right) \\[0.5em] y' = r \left( \sin(\theta) \, \cos(\delta) ~~+~~ \cos(\theta) \, \sin(\delta) \right) \end{cases} \] identités trigonométriques : \[ \begin{cases} x' = r \left( \frac{x}{r} \, \cos(\delta) ~~-~~ \frac{y}{r} \, \sin(\delta) \right) \\[0.5em] y' = r \left( \frac{y}{r} \, \cos(\delta) ~~+~~ \frac{x}{r} \, \sin(\delta) \right) \end{cases} \] distribution et simplification (solution) : \[ \begin{cases} x' = x \, \cos \delta ~~-~~ y \, \sin \delta \\[0.5em] y' = y \, \cos \delta ~~+~~ x \, \sin \delta \end{cases} \] On retrouve ça dans les transformations par matrice : \[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} \quad=\quad \begin{bmatrix} cos \, \delta & -sin \, \delta & 0 \\ sin \, \delta & cos \, \delta & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \quad=\quad \begin{bmatrix} x \, \cos \delta ~-~ y \, \sin \delta \\ x \, \sin \delta ~+~ y \, \cos \delta \\ 1 \end{bmatrix} \] \[ \text{arctan2} = \begin{cases} \arctan \frac{y}{x} & \text{si}\quad & x \gt 0 \\[0.5em] \arctan \frac{y}{x} + \pi & \text{si}\quad & x \lt 0 & y \geq 0 \\[0.5em] \arctan \frac{y}{x} - \pi & \text{si}\quad & x \lt 0 & y \lt 0 \\[0.5em] \frac{\pi}{2} & \text{si}\quad & x = 0 & y \gt 0 \\[0.5em] -\frac{\pi}{2} & \text{si}\quad & x = 0 & y \lt 0 \\[0.5em] \text{indéfini} & \text{si}\quad & x = 0 & y = 0 \end{cases} \]