Title: Calcul / Trigonométrie

Q: cos complexe
A: \(\cos(x)=\frac{e^{ix}+e^{-ix}}{2}\)

Q: sin complexe
A: \(\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}\)

Q: Formule de Moivre
A: \((\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)\)

Q: \(\cos(a+b)\)
A: \(\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)\)

Q: \(\cos(a-b)\)
A: \(\cos(a+b)=\cos(a)\cos(b)+\sin(a)\sin(b)\)

Q: \(\sin(a+b)\)
A: \(\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)\)

Q: \(\sin(a-b)\)
A: \(\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)\)

Q: \(\tan(a+b)\)
A: \(\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}\)

Q: \(\tan(a-b)\)
A: \(\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}\)

Q: \(\cos(2a)\)
A: \(\cos(2a)=\cos^2(a)-\sin^2(a)=1-2\sin^2(a)=2\cos^2(a)-1\)

Q: \(\sin(2a)\)
A: \(\sin(2a)=2\sin(a)\cos(a)\)

Q: \(\tan(2a)\)
A: \(\tan(2a)=\frac{2\tan(a)}{1-\tan^2(a)}\)

Q: \(\cos(a)\cos(b)\)
A: \(\cos(a)\cos(b)=\frac{1}{2}(\cos(a+b)+\cos(a-b))\)

Q: \(\sin(a)\sin(b)\)
A: \(\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))\)

Q: \(\sin(a)\cos(b)\)
A: \(\sin(a)\cos(b)=\frac{1}{2}(\sin(a+b)+\sin(a-b))\)

Q: \(\cos(p)+\cos(q)\)
A: \(2\cos\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)\)

Q: \(\cos(p)-\cos(q)\)
A: \(\cos(p)-\cos(q)=-2\sin\left(\frac{p+q}{2}\right)\sin\left(\frac{p-q}{2}\right)\)

Q: \(\sin(p)+\sin(q)\)
A: \(\sin(p)+\sin(q)=2\sin\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)\)

Q: \(t=\tan\left(\frac{x}{2}\right)\)
A: - \(\cos(x)=\frac{1-t^2}{1+t^2}\)
- \(\sin(x)=\frac{2t}{1+t^2}\)
- \(\tan(x)=\frac{2t}{1-t^2}\)

Q: Somme de arc
A: - \(\arccos(x)+\arcsin(x)=\frac{\pi}{2}\)
- \(\arctan(x)+\arctan(\frac{1}{x})=\frac{\pi}{2}\) pour \(x>0\)