Polar Coordinates

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## Polar Coordinates

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**Polar Coordinates**Lesson 6.3**•**θ r Points on a Plane • Rectangular coordinate system • Represent a point by two distances from the origin • Horizontal dist, Vertical dist • Also possible to represent different ways • Consider using dist from origin, angle formed with positive x-axis (x, y) • (r, θ)**Plot Given Polar Coordinates**• Locate the following**Find Polar Coordinates**• What are the coordinates for the given points? • A • A = • B = • C = • D = • B • D • C**Converting Polar to Rectangular**• Given polar coordinates (r, θ) • Change to rectangular • By trigonometry • x = r cos θy = r sin θ • Try = ( ___, ___ ) • r y θ x**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 = ______ r y θ x**Polar Equations**• States a relationship between all the points (r, θ) that satisfy the equation • Example r = 4 sin θ • Resulting values Note: for (r, θ) It is θ (the 2nd element that is the independent variable θ in degrees**Graphing Polar Equations**• Set Mode on TI calculator • Mode, then Graph => Polar • Note difference of Y= screen**Graphing Polar Equations**• Also best to keepangles in radians • Enter function in Y= screen**Graphing Polar Equations**• Set Zoom to Standard, • then Square**Try These!**• For r = A cos Bθ • Try to determine what affect A and B have • r = 3 sin 2θ • r = 4 cos 3θ • r = 2 + 5 sin 4θ**Polar Form Curves**• Limaçons • r = B ± A cos θ • r = B ± A sin θ**Polar Form Curves**• Cardiods • Limaçons in which a = b • r = a (1 ± cos θ) • r = a (1 ± sin θ)**Polar Form Curves**a • Rose Curves • r = a cos (n θ) • r = a sin (n θ) • If n is odd → n petals • If n is even → 2n petals**Polar Form Curves**• Lemiscates • r2 = a2 cos 2θ • r2 = a2 sin 2θ**Intersection of Polar Curves**• Use all tools at your disposal • Find simultaneous solutions of given systems of equations • Symbolically • Use Solve( ) on calculator • Determine whether the pole (the origin) lies on the two graphs • Graph the curves to look for other points of intersection**Finding Intersections**• Given • Find all intersections**Assignment A**• Lesson 6.3A • Page 384 • Exercises 3 – 29 odd**Area of a Sector of a Circle**• Given a circle with radius = r • Sector of the circle with angle = θ • The area of the sector given by θ r**Area of a Sector of a Region**• Consider a region bounded by r = f(θ) • A small portion (a sector with angle dθ) has area β • dθ α •**Area of a Sector of a Region**• We use an integral to sum the small pie slices β • r = f(θ) α •**Guidelines**• Use the calculator to graph the region • Find smallest value θ = a, and largest value θ = b for the points (r, θ) in the region • Sketch a typical circular sector • Label central angle dθ • Express the area of the sector as • Integrate the expression over the limits from a to b**The ellipse is traced out by 0 < θ < 2π**Find the Area • Given r = 4 + sin θ • Find the area of the region enclosed by the ellipse dθ**Areas of Portions of a Region**• Given r = 4 sin θ and rays θ = 0, θ = π/3 The angle of the rays specifies the limits of the integration**Area of a Single Loop**• Consider r = sin 6θ • Note 12 petals • θ goes from 0 to 2π • One loop goes from0 to π/6**Area Of Intersection**• Note the area that is inside r = 2 sin θand outside r = 1 • Find intersections • Consider sector for a dθ • Must subtract two sectors dθ**Assignment B**• Lesson 6.3 B • Page 384 • Exercises 31 – 53 odd