Lesson 6: Polar, Cylindrical, and Spherical coordinates

Section 9.7Polar, Cylindrical, and Spherical CoordinatesMath 21aFebruary 15, 2008AnnouncementsNo class Monday 2/18. No office hours Tuesday 2/19.Yes office hours Wednesday 2/20 2–4pm SC 323.OutlineWhy different coordinate systems?Polar CoordinatesCylindrical CoordinatesSpherical CoordinatesWhy different coordinate systems?The dimension of space comes from natureThe measurement of space comes from usDifferent coordinate systems are different ways to measurespaceOutlineWhy different coordinate systems?Polar CoordinatesCylindrical CoordinatesSpherical CoordinatesPolar CoordinatesConversion from polar tocartesian (rectangular)x = r cos θy = r sin θrConversion from cartesian toyθpolar:xr =x 2 + y 2xyycos θ =sin θ =tan θ =rrxSolutionWe have√√ −1x =2 cos (5π/4) =2 √ = −12√√ −1y =2 sin (5π/4) =2 √ = −12Example: Polar to RectangularExampleFind the rectangular coordinates of the point with polar√coordinates ( 2, 5π/4).Example: Polar to RectangularExampleFind the rectangular coordinates of the point with polar√coordinates ( 2, 5π/4).SolutionWe have√√ −1x =2 cos (5π/4) =2 √ = −12√√ −1y =2 sin (5π/4) =2 √ = −12Solution√√We have r =3 + 1 =4 = 2, and√31cos θ =sin θ =22This is satisfies by θ = π/6.Example: Rectangular to PolarExampleFind the polar coordinates of the point with rectangular√coordinates ( 3, 1).Example: Rectangular to PolarExampleFind the polar coordinates of the point with rectangular√coordinates ( 3, 1).Solution√√We have r =3 + 1 =4 = 2, and√31cos θ =sin θ =22This is satisfies by θ = π/6.Worksheet #1–#5Document Outline

• Announcements
• Why different coordinate systems?
• Polar Coordinates
• Cylindrical Coordinates
• Spherical Coordinates