2)4.(a, b) = (1,√3)correct5.(a, b) = (√3,1)6.(a, b) = (-1,√3)7.(a, b) = (-2,√3)8.(a, b) = (√3,-1)Explanation:Since the relationship between Cartesiancoordinates and polar coordinates isx=rcosθ ,y=rsinθ ,the pointP(2, π/3) is given in Cartesian co-ordinates byP(2, π/3) =parenleftBig2 cosπ3,2 sinπ3parenrightBig= (1,√3).keywords: polar coordinates, Cartesian coor-dinates01010.0pointsFind a polar equation for the curve givenby the Cartesian equation2y=x2.1.2r= secθtanθ2.r= 2 cscθtanθ3.r= 2 cscθcotθ4.2r= secθcotθ5.2r= cscθcotθ6.r= 2 secθtanθcorrectExplanation:We have to substitute forx, yin2y=x2using the relationsx=rcosθ ,y=rsinθ .In this case the Cartesian equation becomes2rsinθ=r2cos2θ .Consequently, the polar form of the equationisr= 2 secθtanθ.

pacheco (jnp926) – Homework 8 – staron – (52840)
6
011
10.0points
Find a Cartesian equation for the curvegiven by the polar equationr+ 8 sinθ= 0.
01210.0pointsWhich one of the following could be thegraph of the polar curver= 2 cscθ?
1.correct
5.

pacheco (jnp926) – Homework 8 – staron – (52840)
7
6.
Explanation:
As is sometimes the case with polar curves,
it is more convenient to use the relations
x
=
r
cos
θ ,
y
=
r
sin
θ ,
to convert the polar form to Cartesian form.
For then
r
= 2 csc
θ
=
2
sin
θ
becomes
y
= 2 in Cartesian form. Thus the
graph of
r
= 2 sec
θ
is the horizontal line
keywords: polar graph, polar curve, convert
to Cartesian form, line, circle,
013
10.0points
1.
r
= 2 cos
θ
2.
r
= 2 sec
θ
3.
r
= 2 sin
θ
4.
θ
= 2
5.
r
= 2 csc
θ
6.
r
= 2
correct
Explanation:
When the graph of a polar function cannot
be determined directly, it is sometimes more
convenient to use the relations
x
=
r
cos
θ ,
y
=
r
sin
θ ,
to convert the polar form to Cartesian form
and then use standard knowledge of Cartesian
graphs.
This is often the case with special
lines and circles, so let’s look at the six polar
functions listed above.
1.
InCartesianform
r
= 2 sec
θ
=
2
cos
θ
becomes
x
= 2
anditsgraphisaverticalline
totherightoftheorigin
.

pacheco (jnp926) – Homework 8 – staron – (52840)
8