Remember to show all work, to explain all reasoning. Use additional paper if needed.

For this problem, we will find specific terminal points on the unit circle. The lines $y=0,x=0,y=x,y=-x$ subdivide the unit circle into $8$ equal sectors$.$

a$)$ What is the measurement for each of the central angles?

b$)$ Find the intersection of the line $y=x$ and unit circle in the first quadrant, the point highlighted in the diagram above.

c$)$ By this reasoning, what is the terminal point that corresponds to $t=\frac{}{4}?$Now consider the unit circle subdivided into $6$ equal sectors along with an inscribed hexagon.

a$)$ What is the measurement for each the central angles?

b$)$ Find the distance between $(-x,y)$ and $(x,y).$

c$)$ Use the fact that $(-x,y)$ and $(x,y)$ are on the unit circle, and the inscribed hexagon is equilateral and subdivided into equilateral triangles, to solve for $x.$

d$)$ Find the values of the coordinates for the point $(x,y).$

e$)$ By this reasoning, what is the terminal point that corresponds to $t=\frac{}{3}?$Remember to show all work, to explain all reasoning. Use additional paper if needed.

For this problem, we will find specific terminal points on the unit circle. The lines $y=0,x=0,y=x,y=-x$ subdivide the unit circle into $8$ equal sectors$.$

a$)$ What is the measurement for each of the central angles?

b$)$ Find the intersection of the line $y=x$ and unit circle in the first quadrant, the point highlighted in the diagram above.

c$)$ By this reasoning, what is the terminal point that corresponds to $t=\frac{}{4}?$

Now consider the unit circle subdivided into $6$ equal sectors along with an inscribed hexagon.

a$)$ What is the measurement for each the central angles?

b$)$ Find the distance between $(-x,y)$ and $(x,y).$

c$)$ Use the fact that $(-x,y)$ and $(x,y)$ are on the unit circle, and the inscribed hexagon is equilateral and subdivided into equilateral triangles, to solve for $x.$

d$)$ Find the values of the coordinates for the point $(x,y).$

e$)$ By this reasoning, what is the terminal point that corresponds to $t=\frac{}{3}?$