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Polar Coordinates Lesson 6.3 Points on a Plane • Rectangular coordinate system Represent a point by two distances from the origin Horizontal dist, Vertical dist (x, y) • • Also possible to represent different ways • Consider using dist from origin, angle formed (r, θ) with positive x-axis • θ r Plot Given Polar Coordinates • Locate the following A 2, 4 2 B 4, 3 3 C 3, 2 5 D 1, 4 Find Polar Coordinates • What are the coordinates for the given points? • B • A •D •C •A= •B= •C= •D= Converting Polar to Rectangular • Given polar coordinates (r, θ) Change to rectangular r θ x • By trigonometry x = r cos θ y = r sin θ • Try A 2, 4 = ( ___, ___ ) • y Converting Rectangular to Polar • Given a point (x, y) r Convert to (r, θ) θ • By Pythagorean theorem r2 = x2 + • By trigonometry tan • Try this one … for (2, 1) r = ______ θ = ______ 1 y x x y2 • y 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) θ in degrees It is θ (the 2nd element that is the independent variable Graphing Polar Equations • Set Mode on TI calculator Mode, then Graph => Polar • Note difference of Y= screen Graphing Polar Equations • Also best to keep angles in radians • Enter function in Y= screen Graphing Polar Equations • Set Zoom to Standard, then Square Try These! • For r = A cos B θ 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 r 3 5cos r 3 2sin Polar Form Curves • Cardiods Limaçons in which a = b r = a (1 ± cos θ) r = a (1 ± sin θ) r 3 3sin Polar Form Curves • Rose Curves a r = a cos (n θ) r = a sin (n θ) If n is odd → n petals If n is even → 2n petals r 5cos3 r 5sin 4 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 r 4 cos r 4sin • 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 θ r 1 2 • The area of the sector given by A 2 r Area of a Sector of a Region • Consider a region bounded by r = f(θ) • β dθ • α • A small portion (a sector with angle dθ) has 1 2 area A f ( ) d 2 Area of a Sector of a Region • We use an integral to sum the small pie slices β r = f(θ) • • 1 2 A f ( ) d 2 1 2 r d 2 α Guidelines 1. 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 2. Sketch a typical circular sector • Label central angle dθ 1 2 A r 2 3. Express the area of the sector as 4. Integrate the expression over the limits from a to b Find the Area • Given r = 4 + sin θ Find the area of the region enclosed by the ellipse dθ The ellipse is traced out by 0 < θ < 2π 1 2 2 4 sin 0 2 d 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 1 2 /3 16sin 0 2 d Area of a Single Loop • Consider r = sin 6θ Note 12 petals θ goes from 0 to 2π One loop goes from 0 to π/6 1 2 /6 0 2 sin 6 d Area Of Intersection • Note the area that is inside r = 2 sin θ and outside r = 1 dθ • Find intersections 6 • Consider sector for a dθ and 5 6 Must subtract two sectors 1 2 5 / 6 /6 2sin 2 12 d Assignment B • Lesson 6.3 B • Page 384 • Exercises 31 – 53 odd