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 Also possible to represent different ways Consider using dist from origin, angle formed with positive x-axis r θ (x, y) (r, θ)

Plot Given Polar Coordinates Locate the following

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 trigonometry  x = r cos θ y = r sin θ Try = ( ___, ___ ) θ r x y

Converting Rectangular to Polar Given a point (x, y)  Convert to (r, θ) By Pythagorean theorem r 2 = x 2 + y 2 By trigonometry Try this one … for (2, 1)  r = ______  θ = ______ θ r x y

Polar Equations States a relationship between all the points (r, θ) that satisfy the equation Exampler = 4 sin θ  Resulting values θ in degrees Note: for (r, θ) It is θ (the 2 nd element that is the independent variable Note: for (r, θ) It is θ (the 2 nd 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 = 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 Rose Curves  r = a cos (n θ)  r = a sin (n θ)  If n is odd → n petals  If n is even → 2n petals a

Polar Form Curves Lemiscates  r 2 = a 2 cos 2θ  r 2 = a 2 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 dθdθ α β

Area of a Sector of a Region We use an integral to sum the small pie slices α β r = f(θ)

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 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 dθdθ The ellipse is traced out by 0 < θ < 2π

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 from 0 to π/6

Area Of Intersection Note the area that is inside r = 2 sin θ and outside r = 1 Find intersections Consider sector for a dθ  Must subtract two sectors dθdθ

Assignment B Lesson 6.3 B Page 384 Exercises 31 – 53 odd