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

Wierzycaz 2021-02-09 Answered

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 [rcosθ=4 and rsinθ= 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=2secθ and r=3cscθ] are lines " is true or false.

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Expert Answer

delilnaT
Answered 2021-02-10 Author has 94 answers

a)
The polar coordinate [rθ] corresponding to Cartesian coordinate [x y]is given by[r=x2 + y2 andθ=tan1(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=tan1(22)
=tan1(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

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