Exercice : Exercice 27 :

Exprimer chacune des expressions suivantes en fonction de \(sin x\) et \(cos x\).

Question

a. \(sin(x-\frac{\pi}{3})\)

Solution

\(sin(x-\frac{\pi}{3})=sin(x)cos(\frac{\pi}{3})-cos(x)sin(\frac{\pi}{3})=sin(x) \times \frac{1}{2}-cos(x) \times \frac{\sqrt{3}}{2}\)

Question

b. \(\sqrt{2}cos(x +\frac{\pi}{4})\)

Solution

\(\sqrt{2}cos(x +\frac{\pi}{4})=\sqrt{2}[cos(x)cos(\frac{\pi}{4})-sin(x)sin(\frac{\pi}{4})]=\sqrt{2}[\frac{\sqrt{2}}{2}cos(x)-\frac{\sqrt{2}}{2}sin(x)]=\sqrt{2} \times \frac{\sqrt{2}}{2}cos(x)-\sqrt{2} \times \frac{\sqrt{2}}{2}sin(x)=\frac{\sqrt{2}^2}{2}cos(x)-\frac{\sqrt{2}^2}{2}sin(x)=cos x- sin x\)

Question

c.\(\sqrt{2}sin(x -\frac{\pi}{4})\)

Solution

\(\sqrt{2}sin(x -\frac{\pi}{4})=\sqrt{2}[sin(x)cos(\frac{\pi}{4})-cos(x)sin(\frac{\pi}{4})]=\sqrt{2}[\frac{\sqrt{2}}{2}sin(x)-\frac{\sqrt{2}}{2}cos(x)]=\sqrt{2} \times \frac{\sqrt{2}}{2}sin(x)-\sqrt{2} \times \frac{\sqrt{2}}{2}cos(x)=\frac{\sqrt{2}^2}{2}sin(x)-\frac{\sqrt{2}^2}{2}cos(x)=sin x- cos x\)