How to find all the possible solutions for $3 \cos (x) = 0.1$ in the domain of $0 < x < 2\pi$ ? using the unit circle

Question

I am not sure in which way does cos turns in the unit circle. Also how does the amplitude affect the cos graph or the unit of circle? I have found the first solution $x=88.09$ but unable to find the second solution. Please help thank you!

Answer 1

$3 \cos(x) = 0.1$ means $\cos(x) = \frac{1}{30}$. The first solution you found can be noted $r_1$ and corresponds to $\arccos(\frac{1}{30})$. Now notice that $\cos(-x) = \cos(x)$, so the second solution is $r_2 = -r_1$. However, it's not in $[0°,360°]$ range, so you have to add $360°$. So the solutions are $r_1 = \arccos(\frac{1}{30})$ and $r_2 = 360° - r_1$.