Answer: [tex]\bold{b)\quad \dfrac{5}{6},\ \dfrac{25}{6}}[/tex]
Step-by-step explanation:
[tex]12cos\bigg(\dfrac{2\pi}{5}x\bigg)+10=16\\\\\\12cos\bigg(\dfrac{2\pi}{5}x\bigg)=6\\\\\\cos\bigg(\dfrac{2\pi}{5}x\bigg)=\dfrac{1}{2}\\\\\\cos^{-1}\bigg[cos\bigg(\dfrac{2\pi}{5}x\bigg)\bigg]=cos^{-1}\bigg(\dfrac{1}{2}\bigg)[/tex]
[tex]\text{Use the Unit Circle to determine when}\ cos=\dfrac{1}{2}\quad \rightarrow \quad\text{at}\ \dfrac{\pi}{3}\ and \ \dfrac{5\pi}{3}\\\\\\\dfrac{2\pi}{5}x=\dfrac{\pi}{3}\qquad and\qquad \dfrac{2\pi}{5}x=\dfrac{5\pi}{3}\\\\\\\bigg(\dfrac{5}{2\pi}\bigg)\dfrac{2\pi}{5}x=\dfrac{\pi}{3}\bigg(\dfrac{5}{2\pi}\bigg)\quad and\quad \bigg(\dfrac{5}{2\pi}\bigg)\dfrac{2\pi}{5}x=\dfrac{5\pi}{3}\bigg(\dfrac{5}{2\pi}\bigg)\\\\\\.\qquad \qquad \large\boxed{x=\dfrac{5}{6}\qquad and \qquad x=\dfrac{25}{6}}[/tex]
