To determine: a) Whether the statement, " The point with Cartesian coordinates [−2 2] has polar coordinates [bf(22, 3π4) (22,11π4) (22, −5π4) and (−22, −π4)] " is true or false. b) Whether the statement, " the graphs of [rcos⁡θ=4 and rsin⁡θ= −2] intersect exactly once " is true or false. c) Whether the statement, " the graphs of [r=4 and θ=π4] intersect exactly once ", is true or false. d) Whether the statement, " the point [3π2]lies on the graph of[r=3cos⁡ 2 θ] " is true or false. e) Whether the statement, " the graphs of [r=2sec⁡θ and r=3csc⁡θ] are lines " is true or false.

Question

To determine: a) Whether the statement, &quot; The point with Cartesian coordinates [−2 2] has polar coordinates [bf(22, 3π4) (22,11π4) (22, −5π4) and (−22, −π4)] &quot; is true or false. b) Whether the statement, &quot; the graphs of [rcos⁡θ=4 and rsin⁡θ= −2] intersect exactly once &quot; is true or false. c) Whether the statement, &quot; the graphs of [r=4 and θ=π4] intersect exactly once &quot;, is true or false. d) Whether the statement, &quot; the point [3π2]lies on the graph of[r=3cos⁡ 2 θ] &quot; is true or false. e) Whether the statement, &quot; the graphs of [r=2sec⁡θ and r=3csc⁡θ] are lines &quot; is true or false.

delilnaT · Accepted Answer

a) The polar coordinate [rθ] corresponding to Cartesian coordinate [x y]is given by[r=x2 + y2 andθ=tan−1(yx)]. From the given data, [(x, y)=(−2, 2)]. Substitute [(x, y)=(−2, 2)]∈[r=x2 + y2] and obtain the value of r. r=22 + (−2)2 =±8 =± 22 Similary, calculate the value of [θ] as follows. θ=tan−1(−22) =tan−1(−1) =3π4 +πn, n ∈Z For [r=22 and n=0]the polar c in coordinate is[223π4]. For [r=22 and n=2∖]the polar c in coordinate is[(22,11π4)∖]. For [r=22 and n= −2∖]the polar c in coordinate is[(22, −5π4)∖]. For [r=22 and n= −1∖]the polar c in coordinate is[(−22, −π4)∖] Hence the given stament is true. b) Note that the Cartesian coordinates corresponding to polar coordinates

To determine: a) Whether the statement, " The point with Cartesian coordinates \[( -2,\ 2 )\] has polar coordinates \[\bf \left( 2 \sqrt{2},\ \frac{3

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