85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.

r = 2 cos θ and r = 1 + cos θ

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(-4, 4√3)

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.

(-1, -π/3)

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

(x - 1)² + y² = 1

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 2θ; (π/2, π/4)