Polar CoordinatesPoints on a PlanePlot Given Polar CoordinatesFind Polar CoordinatesConverting Polar to RectangularConverting Rectangular to PolarPolar EquationsGraphing Polar EquationsSlide 9Slide 10Try These!Finding dy/dxSlide 13ExampleDefine for CalculatorTry This!AssignmentPolar CoordinatesLesson 10.5Points on a Plane•Rectangular coordinate systemRepresent a point by two distances from the originHorizontal dist, Vertical dist•Also possible to represent different ways•Consider using dist from origin, angle formed with positive x-axis••rθ(x, y)(r, θ)Plot Given Polar Coordinates•Locate the following2,4Ap� �=� �� �33,2Cp� �= -� �� �24,3Bp� �=� �� �51,4Dp� �= -� �� �Find Polar Coordinates•What are the coordinates for the given points?• B• A• C• D• A = • B = • C = • D =Converting Polar to Rectangular•Given polar coordinates (r, θ)Change to rectangular•By trigonometryx = r cos θy = r sin θ •Try = ( ___, ___ )•θrxy2,4Ap� �=� �� �Converting Rectangular to Polar•Given a point (x, y)Convert to (r, θ)•By Pythagorean theorem r2 = x2 + y2•By trigonometry•Try this one … for (2, 1) r = ______θ = ______•θrxy1tanyxq-=Polar Equations•States a relationship between all the points (r, θ) that satisfy the equation•Example r = 4 sin θResulting valuesθ in degreesNote: for (r, θ) It is θ (the 2nd element that is the independent variableNote: for (r, θ) It is θ (the 2nd element that is the independent variableGraphing Polar Equations•Set Mode on TI calculatorMode, then Graph => Polar•Note difference of Y= screenGraphing Polar Equations•Also best to keepangles in radians•Enter function in Y= screenGraphing Polar Equations•Set Zoom to Standard, then SquareTry These!•For r = A cos B θTry to determine what affect A and B have•r = 3 sin 2θ•r = 4 cos 3θ•r = 2 + 5 sin 4θ12Finding dy/dx•We knowr = f(θ) and y = r sin θ and x = r cos θ•Then•And ( ) sin ( ) cosy f x fq q q q= � = �//dy dy ddx dx dqq=13Finding dy/dx•Since•Then //dy dy ddx dx dqq=( ) ( )( ) ( )' sin cos' cos sin' sin cos' cos sinf fdydx f fr rr rq q q qq q q qq qq q� + �=� - �� + �=� - �14Example•Given r = cos 3θFind the slope of the line tangent at (1/2, π/9)dy/dx = ?Evaluate •3sin 3 sin cos3 cos3sin 3 cos cos3 sindydxq q q qq q q q- � + �=- � - �.160292dydx=Define for Calculator•It is possible to define this derivative as a function on your calculator1516Try This!•Find where the tangent line is horizontal for r = 2 cos θ•Find dy/dx•Set equal to 0, solve for θAssignment•Lesson 10.4•Page 736•Exercises 1 – 19 odd, 23 – 26 all•Exercises 69 – 91
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