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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)601110.0pointsFind 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.correct5.
pacheco (jnp926) – Homework 8 – staron – (52840)76.Explanation:As is sometimes the case with polar curves,it is more convenient to use the relationsx=rcosθ ,y=rsinθ ,to convert the polar form to Cartesian form.For thenr= 2 cscθ=2sinθbecomesy= 2 in Cartesian form. Thus thegraph ofr= 2 secθis the horizontal linekeywords: polar graph, polar curve, convertto Cartesian form, line, circle,01310.0points1.r= 2 cosθ2.r= 2 secθ3.r= 2 sinθ4.θ= 25.r= 2 cscθ6.r= 2correctExplanation:When the graph of a polar function cannotbe determined directly, it is sometimes moreconvenient to use the relationsx=rcosθ ,y=rsinθ ,to convert the polar form to Cartesian formand then use standard knowledge of Cartesiangraphs.This is often the case with speciallines and circles, so let’s look at the six polarfunctions listed above.1.InCartesianformr= 2 secθ=2cosθbecomesx= 2anditsgraphisaverticallinetotherightoftheorigin.