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

Please answer all of the following questions (questions 1-7). Show all your work.

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