Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry Chapter 1 Topics in Analytic Geometry
••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 20• rotating the polar axis 5π/4 and then moving forward from the pole 3 units along theterminal side of the angle, or• rotating the polar axis π/4 and then moving backward from the pole 3 units along theextension of the terminal side.Terminal sideP(3,5π/4)5π/4Polar axisP(3,π/4)π/4Terminal sidePolar axisThis suggest that the point (3,5π/4) might also be denoted by (−3,π/4), with minussign serving to indicate that the point is on the extension of the angle’s terminal side ratherthan on the terminal side itself.In general, the terminal side of the angle θ +π is the extension of the terminal side ofθ, we define negative radial coordinates by agreeing thatto be polar coordinates for the same point.(−r,θ) and (r,θ+π)Relationship Between Polar and Rectangular CoordinatesFrequently, it will be useful to superimpose a rectangular xy-coordinate system on top ofa polar coordinate system, making the positive x-axis coincide with the polar axis. Thenevery point P will have both rectangular coordinates (x,y) and polar coordinates (r,θ).yP(r,θ) = P(x,y)ryθxxFrom the above Figure, these coordinates are related by the equationsx = rcosθ, y = rsinθ (2.4)
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 21These equation are well suited for finding x and y when r and θ are known. However, tofind r and θ when x and y are known, we use equationsr 2 = x 2 +y 2 , tanθ = y x(2.5)Example 2.10 Find the rectangular coordinates of the point P whose polar coordinates are(4,2π/3).Solution .........Example 2.11 Find polar coordinates of the point P whose rectangular coordinates are(1,−1).Solution .........Graphs in Polar CoordinatesWe will now consider the problem of graphing equations in r and θ, where θ is assumed tobe measured in radians. Some examples of such equations areExample 2.12 Sketch the graph ofr = 1, θ = π/4, r = θ, r = sinθ, r = cos2θ(a) r = 1 (b) θ = π/4Solution .........Example 2.13 Sketch the graph of r = 2cosθ in polar coordinates by plotting points.Solution .........Example 2.14 Sketch the graph of r = cos2θ in polar coordinates.Solution Instead of plotting points, we will use the graph of r = cos2θ in rectangularcoordinates to visualize how the polar graph of this equation is generated. This curve iscalled a four-petal rose.
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 21These equation are well suited for f<strong>in</strong>d<strong>in</strong>g x and y when r and θ are known. However, tof<strong>in</strong>d r and θ when x and y are known, we use equationsr 2 = x 2 +y 2 , tanθ = y x(2.5)Example 2.10 F<strong>in</strong>d the rectangular coord<strong>in</strong>ates of the po<strong>in</strong>t P whose polar coord<strong>in</strong>ates are(4,2π/3).Solution .........Example 2.11 F<strong>in</strong>d polar coord<strong>in</strong>ates of the po<strong>in</strong>t P whose rectangular coord<strong>in</strong>ates are(1,−1).Solution .........Graphs <strong>in</strong> Polar Coord<strong>in</strong>atesWe will now consider the problem of graph<strong>in</strong>g equations <strong>in</strong> r and θ, where θ is assumed tobe measured <strong>in</strong> radians. Some examples of such equations areExample 2.12 Sketch the graph ofr = 1, θ = π/4, r = θ, r = s<strong>in</strong>θ, r = cos2θ(a) r = 1 (b) θ = π/4Solution .........Example 2.13 Sketch the graph of r = 2cosθ <strong>in</strong> polar coord<strong>in</strong>ates by plott<strong>in</strong>g po<strong>in</strong>ts.Solution .........Example 2.14 Sketch the graph of r = cos2θ <strong>in</strong> polar coord<strong>in</strong>ates.Solution Instead of plott<strong>in</strong>g po<strong>in</strong>ts, we will use the graph of r = cos2θ <strong>in</strong> rectangularcoord<strong>in</strong>ates to visualize how the polar graph of this equation is generated. This curve iscalled a four-petal rose.
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