Exercice : Exercice 5

1.Simplifier chacune des expressions suivantes :

Question

a.\(cos(x-\pi)\)

Solution

\(cos(x-\pi)=-cos(x)\)

Question

b.\(sin(x-\frac{\pi}{2})\)

Solution

\(sin(x-\frac{\pi}{2})=-cos(x)\)

Question

c.\(sin(x+\frac{\pi}{2})\)

Solution

\(sin(x+\frac{\pi}{2})=cos(x)\)

Question

d.\(cos(x+\frac{\pi}{2})\)

Solution

\(cos(x+\frac{\pi}{2})=-sin(x)\)

2.A l'aide de la relation :

\(tan x= \frac{sin x}{cos x}\) où \(x\ne \frac{\pi}{2}+k\pi\)

simplifier les expressions suivantes :

Question

a.\(tan(x+\pi)\)

Solution

\(tan (x+\pi)= \frac{sin(x+\pi)}{cos(x+\pi)}\)

\(sin(x+\pi)=-sin(x)\)

\(sin(x+\pi)=-cos(x)\)

\(\iff tan (x+\pi)= \frac{-sin(x)}{-cos(x)}\)

\(\iff tan (x+\pi)= \frac{sin(x)}{cos(x)}\)

\(\iff tan (x+\pi)= tan(x)\)

Question

b.\(tan(\frac{\pi}{2}-x)\)

Solution

\(tan(\frac{\pi}{2}-x)= \frac{sin(\frac{\pi}{2}-x)}{cos(\frac{\pi}{2}-x)}\)

\(sin(\frac{\pi}{2}-x)=cos(x)\)

\(cos(\frac{\pi}{2}-x)=sin(x)\)

\(\iff tan(\frac{\pi}{2}-x)= \frac{cos(x)}{sin(x)}\)

\(\iff tan(\frac{\pi}{2}-x)=\frac{1}{\frac{sin(x)}{cos(x)}}\)

\(\iff tan(\frac{\pi}{2}-x)= \frac{1}{tan(x)}\)