Graphing Polar Equations Line Equation (when the equation is EQUAL to an answer) θ = b r cos θ = b r sin θ = b Circle Equation (when the radius is given or needs to be solved) r = 3 r = + 2a cos θ where a is the center r = + 2a sin θ where a is the center 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 9 Graph 𝜽= 𝝅 𝟔 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 10 Graph 𝒓𝐬𝐢𝐧𝛉=−𝟑 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph 𝒓𝐜𝐨𝐬𝛉=𝟐 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 11 Graph r = 3 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 12 Graph 𝒓=−𝟔𝐬𝐢𝐧𝛉 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph 𝒓=𝟖𝐜𝐨𝐬𝛉 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Calculus AP/Dual, Revised ©2013 Graphing Polar Section 11.5 Calculus AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Forms Introduced Polar Coordinates: (r, θ) Rectangular/Standard Form: (x, y) 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Goes through the origin Radius measured from the origin Type Graph Rules Equation Example Graphic Organizer Line Goes through the origin 𝜽=𝒃 𝒓 𝐜𝐨𝐬 𝜽=𝒃 𝒓 𝐬𝐢𝐧 𝜽=𝒃 𝜽= 𝝅 𝟔 , 𝐫𝐜𝐨𝐬𝜽=𝟐, 𝐫𝐬𝐢𝐧𝜽=−𝟑 Circle Radius measured from the origin 𝒓=𝒂 𝒓=±𝟐𝒂 𝐜𝐨𝐬 𝜽 𝒓=±𝟐𝒂 𝐬𝐢𝐧 𝜽 𝒓=𝟑, 𝐫=𝟖 𝐜𝐨𝐬 𝜽, 𝐫=−𝟔 𝐬𝐢𝐧 𝜽 Cardioid (Heart-Shaped) 𝒂 𝒃 =𝟏 Generally, a = b 𝒓=𝒂±𝒃 𝐜𝐨𝐬 𝛉 𝒓=𝒂±𝒃 𝒔𝒊𝒏 𝛉 𝒓=𝟐+𝟐 𝐬𝐢𝐧 𝜽, 𝒓=𝟑 −𝟑 𝐜𝐨𝐬 𝜽 Limaçon (Inner Loop) 𝒂 𝒃 <𝟏 “Less = Loop” 𝒓=𝒂±𝒃 𝐜𝐨𝐬 𝛉 𝒓=𝒂±𝒃 𝒔𝒊𝒏 𝛉 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽, 𝒓=𝟑 −𝟑 𝐜𝐨𝐬 𝜽 Limaçon (Dimpled) 𝟏< 𝒂 𝒃 <𝟐 “Dimple Between” 𝒓=𝒂±𝒃 𝐜𝐨𝐬 𝛉 𝒓=𝒂±𝒃 𝒔𝒊𝒏 𝛉 𝒓=𝟐+𝟑 𝐬𝐢𝐧 𝜽, 𝒓=𝟑 −𝟑 𝐜𝐨𝐬 𝜽 Limaçon (Convex) 𝒂 𝒃 >𝟐 “Dimple Between” 𝒓=𝒂±𝒃 𝐜𝐨𝐬 𝛉 𝒓=𝒂±𝒃 𝒔𝒊𝒏 𝛉 𝒓=𝟑+𝟏 𝐬𝐢𝐧 𝜽, 𝒓=𝟑 −𝟑 𝐜𝐨𝐬 𝜽 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Graphic Organizer Type Graph Rules Equation Example Rose (cos nθ) when n = odd n is odd, n pedals Cos: 1st pedal is at 0 radian 𝒓=𝒂 𝐜𝐨𝐬 𝒏𝜽 𝒓=𝟐 𝐜𝐨𝐬 𝟑𝜽 Rose (cos nθ) when n = even n is even, 2n pedals Cos: 1st pedal is at 0 radian 𝒓=𝒂 𝐜𝐨𝐬 𝒏𝜽 𝒓=𝟑 𝐜𝐨𝐬 𝟐𝜽 Rose (sin nθ) when n = odd n is odd, n pedals Sin: 1st pedal is at 𝝅 𝟐𝒏 radian 𝒓=𝒂 𝐬𝐢𝐧 𝒏𝜽 𝒓=𝟐 𝐬𝐢𝐧 𝟑𝜽 Rose (sin nθ) when n = even n is even, 2n pedals Sin: 1st pedal is at 𝝅 𝟐𝒏 radian 𝒓=𝒂 𝐬𝐢𝐧 𝒏𝜽 𝒓=𝟑 𝐬𝐢𝐧 𝟐𝜽 Lemniscates (cos 2θ) n is odd, n pedals Cos: loops are at 0 and π 𝒓 𝟐 = 𝒂 𝟐 𝐜𝐨𝐬 𝟐𝛉 𝒓 𝟐 =𝟗 𝐜𝐨𝐬 𝟐𝛉 Lemniscates (sin 2θ) n is even, n pedals Sin: loops are at π/4 and 5π/4 𝒓 𝟐 = 𝒂 𝟐 𝐬𝐢𝐧 𝟐𝛉 𝒓 𝟐 =𝟗 𝐬𝐢𝐧 𝟐𝛉 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Graphic Organizer Type Graph Rules Equation Example Archimedes Spiral when a = positive a = positive CLOCKWISE 𝒓=𝒂 𝜽 𝒓=𝟐𝜽 Archimedes Spiral when a = negative a = negative COUNTER CLOCKWISE 𝒓=−𝒂 𝜽 𝒓=−𝟐𝜽 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 1 Graph r = 3 + 3 sin θ r 3 6 r θ π/2 π 3π/2 2π r θ π/2 π 3π/2 2π 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph r = 2 + 2 cos θ r θ π/2 π 3π/2 2π 12/4/2018 5:03 PM π/2 π 3π/2 2π 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 2 Graph r = 2 + 4 sin θ r θ π/2 π 3π/2 2π r 2 6 –2 π/2 π 3π/2 2π r 2 6 –2 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 3 Graph r = 4 + 2 cos θ r θ π/2 π 3π/2 2π r 6 4 2 π/2 π 3π/2 2π r 6 4 2 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 4 Graph r = 4 + 3 cos θ r θ π/2 π 3π/2 2π r 7 4 1 π/2 π 3π/2 2π r 7 4 1 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph r = 1 – 2 cos θ r θ π/2 π 3π/2 2π r 1 –3 π/2 π 3π/2 2π r 1 –3 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 5 Graph r = 4 cos 3θ r θ 2π/3 π 4π/3 2π r 4 –4 2π/3 π 4π/3 2π r 4 –4 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 6 Graph r = 5 cos 2θ 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph r = 4 cos 2θ 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 7 Graph r2 = 16 cos 2θ r θ π/2 π 3π/2 2π r 4 –4 π/2 π 3π/2 2π r 4 –4 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 8 Graph r2 = 16 sin 2θ r θ π/4 π 5π/4 2π r 4 –4 π/4 π 5π/4 2π r 4 –4 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph r2 = 9 sin 2θ 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Example 9 Graph r = 2θ r θ π/2 π 3π/2 2π r π 2π 3π 4π π/2 π 3π/2 2π r π 2π 3π 4π 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Your Turn Graph r = –2θ 12/4/2018 5:03 PM 11.5 - Polar Coordinates

Assignment Worksheet 2 12/4/2018 5:03 PM 11.5 - Polar Coordinates