Exercice : Somme 5

1. Calculer :

Question

a. \(\pi-\frac{2\pi}{5}\)

Solution

\(\pi-\frac{2\pi}{5}=\frac{5\pi}{5}-\frac{2\pi}{5}=\frac{3\pi}{5}\)

Question

b. \(\pi-\frac{\pi}{5}\)

Solution

\(\pi-\frac{\pi}{5}=\frac{5\pi}{5}-\frac{\pi}{5}=\frac{4\pi}{5}\)

Question

c. \(2\pi-\frac{\pi}{5}\)

Solution

\(2\pi-\frac{\pi}{5}=\frac{10\pi}{5}-\frac{\pi}{5}=\frac{9\pi}{5}\)

Question

d. \(2\pi-\frac{2\pi}{5}\)

Solution

\(2\pi-\frac{\pi}{5}=\frac{10\pi}{5}-\frac{2\pi}{5}=\frac{8\pi}{5}\)

Question

e. \(\pi+\frac{\pi}{5}\)

Solution

\(\pi+\frac{\pi}{5}=\frac{5\pi}{5}+\frac{\pi}{5}=\frac{6\pi}{5}\)

2. En déduire la valeur exacte de :

Question

•\( A=cos\frac{\pi}{5}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}+cos\frac{9\pi}{5}\)

Solution

\(A=cos\frac{\pi}{5}+cos\frac{2\pi}{5}+cos(\pi-\frac{2\pi}{5})+cos(2\pi-\frac{\pi}{5})\)

\(\iff A=cos\frac{\pi}{5}+cos\frac{2\pi}{5}-cos(\frac{2\pi}{5})+cos(\frac{\pi}{5})\)

\(\iff A=2cos\frac{\pi}{5}\)

Question

\(• B=sin\frac{2\pi}{5}+sin\frac{4\pi}{5}+sin\frac{6\pi}{8}+sin\frac{8\pi}{5}\)

Solution

\(• B=sin\frac{2\pi}{5}+sin(\pi-\frac{\pi}{5})+sin(\pi+\frac{\pi}{5})+sin(2\pi-\frac{2\pi}{5})\)

\(\iff B=sin\frac{2\pi}{5}+sin(\frac{\pi}{5})-sin(\frac{\pi}{5})-sin(\frac{2\pi}{5})\)

\(\iff B=0\)