Chapitre VII Trigonométrie

• Déterminons la mesure principale de l'angle de mesure \(\frac{99\pi}{8}\)

\(\frac{\frac{99\pi}{8}}{2\pi}=\frac{\frac{99\pi}{8}}{\frac{2\pi}{1}}\)

\(\iff \frac{\frac{99\pi}{8}}{2\pi}=\frac{99\pi}{8} \times \frac{1}{2\pi}\)

\(\iff \frac{\frac{99\pi}{8}}{2\pi}=\frac{99}{16}\)

\(\frac{96}{16} \le \frac{99}{16} \le \frac{112}{16}\)

\(\iff 6 \le \frac{99}{16} \le 7\)

• \(\frac{99\pi}{8}- 6 \times 2\pi=\frac{99\pi}{8}- 12\pi\)

\(\iff \frac{99\pi}{8}- 6 \times 2\pi=\frac{99\pi}{8}- \frac{96\pi}{8}\)

\(\iff \frac{99\pi}{8}- 6 \times 2\pi=\frac{3\pi}{8}\)

• \(\frac{99\pi}{8}- 7 \times 2\pi=\frac{99\pi}{8}- 14\pi\)

\(\iff \frac{99\pi}{8}- 7 \times 2\pi=\frac{99\pi}{8}- \frac{112\pi}{8}\)

\(\iff \frac{99\pi}{8}- 7 \times 2\pi=\frac{-13\pi}{8}\)

La mesure principale de \(\frac{99\pi}{8}\) est donc \(\frac{3\pi}{8}\)

• Déterminons la mesure principale de l'angle de mesure \(\frac{97\pi}{8}\)

\(\frac{\frac{97\pi}{8}}{2\pi}=\frac{\frac{97\pi}{8}}{\frac{2\pi}{1}}\)

\(\iff \frac{\frac{97\pi}{8}}{2\pi}=\frac{97\pi}{8} \times \frac{1}{2\pi}\)

\(\iff \frac{\frac{97\pi}{8}}{2\pi}=\frac{97}{16}\)

\(\frac{96}{16} \le \frac{97}{16} \le \frac{112}{16}\)

\(\iff 6 \le \frac{97}{16} \le 7\)

• \(\frac{97\pi}{8}- 6 \times 2\pi=\frac{97\pi}{8}- 12\pi\)

\(\iff \frac{97\pi}{8}- 6 \times 2\pi=\frac{97\pi}{8}- \frac{96\pi}{8}\)

\(\iff \frac{97\pi}{8}- 6 \times 2\pi=\frac{\pi}{8}\)

• \(\frac{97\pi}{8}- 7 \times 2\pi=\frac{97\pi}{8}- 14\pi\)

\(\iff \frac{97\pi}{8}- 7 \times 2\pi=\frac{97\pi}{8}- \frac{112\pi}{8}\)

\(\iff \frac{97\pi}{8}- 7 \times 2\pi=\frac{-15\pi}{8}\)

La mesure principale de \(\frac{99\pi}{8}\) est donc \(\frac{\pi}{8}\)

\(\frac{\pi}{2}-\frac{3\pi}{8}=\frac{4\pi}{8}-\frac{3\pi}{8}=\frac{\pi}{8}\)

Les deux angles sont donc distincts de \(\frac{\pi}{2}\)

or \(sin(\frac{\pi}{2}-x)=cos(x)\) donc la proposition est vraie.