ANSWERS
x=0.5 in Quadrant I.
θ = 60 degrees
x=-0.8 in Quadrant II.
θ = 143.13 degrees
x=-0.1 in Quadrant III.
θ = 264.26 degrees
x=0.8 in Quadrant IV.
θ = 323.13 degrees
EXPLANATION
For all the angles we know the length of the adjacent side of the triangle and the length of the hypotenuse - which is the radius of the circle.
For the first triangle in red, the angle is:
[tex]\begin{gathered} \cos \theta=\frac{0.5}{1} \\ \cos \theta=0.5 \\ \theta=\cos ^{-1}0.5 \\ \theta=60º \end{gathered}[/tex]For the second triangle, in green, we can find the supplementary angle of θ:
[tex]\begin{gathered} \cos (180º-\theta)=\frac{0.8}{1} \\ 180º-\theta=\cos ^{-1}0.8 \\ 180º-\theta=36.87º \\ \theta=180º-36.87º \\ \theta=143.13º \end{gathered}[/tex]For the third triangle, in light-blue, the angle we'll find is (θ - 180º):
[tex]\begin{gathered} \cos (\theta-180º)=\frac{0.1}{1} \\ \theta-180º=\cos ^{-1}0.1 \\ \theta=180º+84.26º \\ \theta=264.26º \end{gathered}[/tex]And for the last triangle, in pink, the angle we'll find from the triangle is (360º-θ):
[tex]\begin{gathered} \cos (360º-\theta)=\frac{0.8}{1} \\ 360º-\theta=\cos ^{-1}0.8 \\ \theta=360º-36.87º \\ \theta=323.13º \end{gathered}[/tex]
