# Please answer all of the following questions (questions 1-7). Show all your work.) Find the Cartesian coordinates of the given polar coordinates….

1.) Find the Cartesian coordinates of the given polar coordinates. Then plot the point.

(a)    (4, π)

(b)    (4, −2π/3)

(c)    (−4, 3π/4)

2.) The Cartesian coordinates of a point are given.

(a)    (4, −4)

(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.

(r, θ) =

(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.

(r, θ) =

(b)    (−1, 3)

(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.

(r, θ) =

(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.

(r, θ) =

3.) Find two other pairs of polar coordinates of the given polar coordinate, one with r > 0

and one with r < 0.

Then plot the point.

(a)    (2, 7π/4)

(r, θ) =      (r > 0)(r, θ) =      (r < 0)

(b)    (−5, π/3)

(r, θ) =      (r > 0)(r, θ) =      (r < 0)

(c)    (2, −3)

(r, θ) =      (r > 0)(r, θ) =      (r < 0)

4.) Find the distance between the points with polar coordinates (4, π/3) and (8, 2π/3).

5.) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

r ≥ 5,    π ≤ θ ≤ 2pi

6.) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

1 < r ≤ 2,    5π/6 ≤ θ ≤ 7π/6

7.) Find the slope of the tangent line to the given polar curve at the point specified by the value of θ.

r = 6 sin(θ),    θ = π/6