POLAR COORDINATES Section 11 5 Calculus APDual Revised

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POLAR COORDINATES Section 11. 5 Calculus AP/Dual, Revised © 2013 viet. dang@humble. k 12.

POLAR COORDINATES Section 11. 5 Calculus AP/Dual, Revised © 2013 viet. dang@humble. k 12. tx. us 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 1

WHAT IS POLAR FORM? A. The polar form of a complex number is another

WHAT IS POLAR FORM? A. The polar form of a complex number is another way to represent a complex number. B. Polar Coordinates are two values that locate a point on a plane by its distance from a fixed pole and its angle from a fixed line passing through the pole 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 2

FORMS INTRODUCED A. Polar Coordinates: (r, θ) B. Rectangular/Standard Form: (x, y) 3/2/2021 4:

FORMS INTRODUCED A. Polar Coordinates: (r, θ) B. Rectangular/Standard Form: (x, y) 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 3

USING TRIGONOMETRY A. x = r cos θ B. y = r sin θ

USING TRIGONOMETRY A. x = r cos θ B. y = r sin θ 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 4

REVIEW A. Plot (3, 60°) on a regular graph: 3/2/2021 4: 48 AM 11.

REVIEW A. Plot (3, 60°) on a regular graph: 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 5

EXAMPLE 1 Plot (3, 60°) on a polar graph. 3/2/2021 4: 48 AM 11.

EXAMPLE 1 Plot (3, 60°) on a polar graph. 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 6

EXAMPLE 2 Plot (3, – 5π/6) on a polar graph. 3/2/2021 4: 48 AM

EXAMPLE 2 Plot (3, – 5π/6) on a polar graph. 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 7

EXAMPLE 3 Plot (– 3, –π/6) on a polar graph. 3/2/2021 4: 48 AM

EXAMPLE 3 Plot (– 3, –π/6) on a polar graph. 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 8

YOUR TURN Plot A(– 2, 5π/4) and B(– 1, – 3π/4) on a polar

YOUR TURN Plot A(– 2, 5π/4) and B(– 1, – 3π/4) on a polar graph. 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 9

RECALL Plot (3, 60°) on a polar graph. Are there other ways of labeling

RECALL Plot (3, 60°) on a polar graph. Are there other ways of labeling this point? 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 10

IDENTIFYING MULTIPLE REPRESENTATIONS OF POINTS A. With the argument or the angle 1. Add

IDENTIFYING MULTIPLE REPRESENTATIONS OF POINTS A. With the argument or the angle 1. Add and/or subtract multiples of 2π or 360° B. With the moduli or the “radius” 1. With a negative length, add and/or subtract multiples of π or 180° 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 11

EXAMPLE 4 Plot the point of (3, – 3π/4) on a polar graph. Then,

EXAMPLE 4 Plot the point of (3, – 3π/4) on a polar graph. Then, find three additional polar representations of this graph from (– 2π, 2π) 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 12

YOUR TURN Plot the point of (– 5, – 2π/3) on a polar graph.

YOUR TURN Plot the point of (– 5, – 2π/3) on a polar graph. Then, find three additional polar representations of this graph from (– 2π, 2π) 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 13

CONVERSIONS A. Polar to Rectangular Conversions 1. x = r cos θ 2. y

CONVERSIONS A. Polar to Rectangular Conversions 1. x = r cos θ 2. y = r sin θ B. Rectangular to Polar Conversions 1. r 2 = x 2 + y 2 2. tan θ = y/x (use the inverse rules if using the calculator) 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 14

EXAMPLE 5 Convert (4, π/6) to rectangular form 3/2/2021 4: 48 AM 11. 5

EXAMPLE 5 Convert (4, π/6) to rectangular form 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 15

YOUR TURN Convert (4, 5π/3) to rectangular form 3/2/2021 4: 48 AM 11. 5

YOUR TURN Convert (4, 5π/3) to rectangular form 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 16

EXAMPLE 6 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 17

EXAMPLE 6 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 17

EXAMPLE 7 Convert (0, 2) to polar form 3/2/2021 4: 48 AM 11. 5

EXAMPLE 7 Convert (0, 2) to polar form 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 18

EXAMPLE 8 Convert (– 5, 12) to polar form 3/2/2021 4: 48 AM 11.

EXAMPLE 8 Convert (– 5, 12) to polar form 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 19

YOUR TURN Convert (– 3, – 5) to polar form 3/2/2021 4: 48 AM

YOUR TURN Convert (– 3, – 5) to polar form 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 20

GRAPHING POLAR EQUATIONS A. Line Equation (when the equation is EQUAL to an answer)

GRAPHING POLAR EQUATIONS A. Line Equation (when the equation is EQUAL to an answer) 1. θ = b 2. r cos θ = b 3. r sin θ = b B. Circle Equation (when the radius is given or needs to be solved) 1. r = 3 2. r = + 2 a cos θ where a is the center 3. r = + 2 a sin θ where a is the center 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 21

EXAMPLE 9 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 22

EXAMPLE 9 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 22

EXAMPLE 10 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 23

EXAMPLE 10 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 23

YOUR TURN 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 24

YOUR TURN 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 24

EXAMPLE 11 Graph r = 3 3/2/2021 4: 48 AM 11. 5 - Polar

EXAMPLE 11 Graph r = 3 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 25

EXAMPLE 12 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 26

EXAMPLE 12 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 26

YOUR TURN 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 27

YOUR TURN 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 27

ASSIGNMENT Worksheet 1 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 28

ASSIGNMENT Worksheet 1 3/2/2021 4: 48 AM 11. 5 - Polar Coordinates 28

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