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Looking for an English version of this section?
This section is in French because the contents are in French. If you prefer the English version of this section, click on the link below.
However, the contents in the French and the English sections are not necessarily the same!• Contents for mathematics in English •
However, the contents in the French and the English sections are not necessarily the same!
Quelques formules utiles
Fonction cosinus
Quelques propriétés de la fonction cosinus (cos) :
\(\displaystyle cos(-x)=cos(x)\)
\(\displaystyle cos(\pi+x)=-cos(x)\)
\(\displaystyle cos(\pi-x)=-cos(x)\)
\(\displaystyle cos(\frac{\pi}{2}+x)=-sin(x)\)
\(\displaystyle cos(\frac{\pi}{2}-x)=sin(x)\)
\(\displaystyle cos(x+y)=cos(x)*cos(y)−sin(x)*sin(y)\)
\(\displaystyle cos(x-y)=cos(x)*cos(y)+sin(x)*sin(y)\)
\(\displaystyle \begin{aligned}cos(2x)&=cos^2(x)-sin^2(x) \\
&=1-2*sin^2(x) \\
&=2*cos^2(x)-1\end{aligned}\)
\(\displaystyle \begin{aligned}cos(3x)&=cos(x)*\left(1-4*sin^2(x)\right) \\
&=cos(x)*\left(4*cos^2(x)-3\right)\end{aligned}\)
\(\displaystyle cos^2(\frac{x}{2})=\frac{1+cos(x)}{2}\)
\(\displaystyle cos(x)+cos(y)=2*cos\left(\frac{x+y}{2}\right)*cos\left(\frac{x-y}{2}\right)\)
\(\displaystyle cos(x)-cos(y)=-2*sin\left(\frac{x+y}{2}\right)*sin\left(\frac{x-y}{2}\right)\)
\(\displaystyle cos(x)*cos(y)=\frac{1}{2}\left(cos(x+y)+cos(x-y)\right)\)
\(\displaystyle cos(x)*sin(y)=\frac{1}{2}\left(sin(x+y)-sin(x-y)\right)\)
Fonction sinus
Quelques propriétés de la fonction sinus (sin) :
\(\displaystyle sin(-x)=-sin(x)\)
\(\displaystyle sin(\pi+x)=-sin(x)\)
\(\displaystyle sin(\pi-x)=sin(x)\)
\(\displaystyle sin(\frac{\pi}{2}+x)=cos(x)\)
\(\displaystyle sin(\frac{\pi}{2}-x)=cos(x)\)
\(\displaystyle sin(x+y)=sin(x)*cos(y)+cos(x)*sin(y)\)
\(\displaystyle sin(x-y)=sin(x)*cos(y)-cos(x)*sin(y)\)
\(\displaystyle sin(2x)=2*sin(x)*cos(x)\)
\(\displaystyle \begin{aligned}sin(3x)&=sin(x)\left(4*cos^2(x)-1\right) \\
&=sin(x)*\left(3-4*sin^2(x)\right)\end{aligned}\)
\(\displaystyle sin^2(\frac{x}{2})=\frac{1-cos(x)}{2}\)
\(\displaystyle sin(x)+sin(y)=2*sin\left(\frac{x+y}{2}\right)*cos\left(\frac{x-y}{2}\right)\)
\(\displaystyle sin(x)-sin(y)=2*cos\left(\frac{x+y}{2}\right)*sin\left(\frac{x-y}{2}\right)\)
\(\displaystyle sin(x)*sin(y)=\frac{1}{2}\left(-cos(x+y)+cos(x-y)\right)\)
\(\displaystyle sin(x)*cos(y)=\frac{1}{2}\left(sin(x+y)+sin(x-y)\right)\)
Fonction tangente
Quelques propriétés de la fonction tangente (tan) :
\(\displaystyle tan(-x)=-tan(x)\)
\(\displaystyle tan(\pi+x)=tan(x)\)
\(\displaystyle tan(\pi-x)=-tan(x)\)
\(\displaystyle tan(\frac{\pi}{2}+x)=-cotan(x)\)
\(\displaystyle tan(\frac{\pi}{2}-x)=cotan(x)\)
\(\displaystyle tan(x+y)=\frac{tan(x)+tan(y)}{1-tan(x)*tan(y)}\)
\(\displaystyle tan(x-y)=\frac{tan(x)-tan(y)}{1+tan(x)*tan(y)}\)
\(\displaystyle tan(2x)=\frac{2*tan(x)}{1-tan^2(x)}\)
\(\displaystyle tan(3x)=\frac{tan(x)\left(3-tan^2(x)\right)}{1-3*tan^2(x)}\)
\(\displaystyle tan^2(\frac{x}{2})=\frac{1-cos(x)}{1+cos(x)}\)
\(\displaystyle \begin{aligned}tan(\frac{x}{2})&=\frac{1-cos(x)}{sin(x)} \\
&=\frac{sin(x)}{1+cos(x)}\end{aligned}\)
\(\displaystyle tan(x)+tan(y)=\frac{sin(x+y)}{cos(x)*cos(y)}\)
\(\displaystyle tan(x)-tan(y)=\frac{sin(x-y)}{cos(x)*cos(y)}\)
Si vous avez des commentaires ou des questions sur les formules de trigonométrie, vous pouvez venir en discuter sur le forum : Forums de discussion.
DÉFI : Si vous voulez vous entrainer à démontrer n'importe lesquelles de ces formules, vous pouvez venir proposer une démonstration sur le forum : Forums de discussion.
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