Calculus BC Polar Coordinates

Polar Coordinates

Convert the polar equation to Cartesian coordinates : r = 3tanθ

• Apply definitions for r and θ
• √{x² + y²} = 3tan(tan ^{− 1}([y/x])
• Reduce tangent and arctangent function
• √{x² + y²} = 3[y/x]

√{x² + y²} = 3[y/x] (√{x² + y²} )[x/y] = 3 [(x√{x² + y²} )/y] − 3 = 0

Convert the polar equation to Cartesian coordinates : r³ = − 7cosθ

• Apply definitions for r and θ
• (√{x² + y²} )³ = − 7sin(tan ^{− 1}([y/x]))
• Use trig identity
• (√{x² + y²} )³ = − 7([x/(√{x² + y²} )])
• Isolate for Cartesian form
• (√{x² + y²} )³ = − 7([x/(√{x² + y²} )])
• (√{x² + y²} )³([(√{x² + y²} )/y]) = − 7

[((√{x² + y²} )²(√{x² + y²} )²)/x] = − 7[((x² + y²)(x² + y²))/x] = − 7[((x² + y²)²)/x] + 7 = 0

Convert the polar equation to Cartesian coordinates : r = cosθsinθ

• Apply definitions for r and θ
• √{x² + y²} = cos(tan ^{− 1}([y/x]))sin(tan ^{− 1}([y/x]))
• Apply trig identities
• √{x² + y²} = ([x/(√{x² + y²} )])([y/(√{x² + y²} )])
• √{x² + y²} = [xy/(x² + y²)]
• √{x² + y²} ([(x² + y²)/xy]) = 1
• [(³√{x² + y²})/xy] = 1

[(³√{x² + y²})/xy] − 1 = 0

Convert the polar equation to Cartesian coordinates : r = secθ

• Apply definitions for r and θ
• √{x² + y²} = sec(tan ^{− 1}([y/x]))
• Apply trig identities
• √{x² + y²} = [1/(cos(tan ^{− 1}([y/x])))]
• √{x² + y²} = [1/([x/(√{x² + y²} )])]

√{x² + y²} ([x/(√{x² + y²} )]) = 1x = 1x − 1 = 0

Find the polar representation of [(x²)/25] + y² = 1

• Apply definitions
• [((rcosθ)²)/25] + (rsinθ)² = 1
• Expand
• [(r²cos²θ)/25] + [(r²sin²θ)/1] = 1
• [(r²cos²θ)/25] + [(25(r²sin²θ))/25] = 1
• [(r²cos²θ+ 25(r²sin²θ))/25] = 1
• Simplify
• [(r²(cos²θ+ 25r²sin²θ))/25] = 1
• r² = [25/(cos²θ+ 25r²sin²θ)]

r = √{[25/(cos²θ+ 25r²sin²θ)]}

Find the polar representation of x + [y/36] − 1 = 3

• Apply definitions
• rcosθ+ [((rsinθ))/36] − 1 = 3
• Isolate and expand
• rcosθ+ [((rsinθ))/36] = 4
• [(36(rcosθ) + rsinθ)/36] = 4
• Simplify
• r(36cosθ+ sinθ) = 144

r = [144/(rcosθ+ rsinθ)]

Find the polar representation of 3x + 5y = 17

• Apply definitions
• 3(rcosθ) + 5(rsinθ) = 17
• Simplify
• r(3cosθ+ 5sinθ) = 17

r = [17/(3cosθ+ 5sinθ)]

Find the polar representation of [y/x] − 16y = 9

• Apply definitions
• [(rsinθ)/(rcosθ)] − 16( rsinθ ) = 9
• Simplify

tanθ− 16( rsinθ ) = 9 − 16( rsinθ ) = 9 − tanθr = [(9 − tanθ)/( − 16sinθ)]

Find the polar representation of [y/4] + x − 2 = 0

• Apply definitions
• [(rsinθ)/4] + rcosθ− 2 = 0
• Isolate and simplify
• [(rsinθ)/4] + rcosθ = 2
• r([(sinθ)/4] + cosθ) = 2
• r([(sinθ+ 4cosθ)/4]) = 2

r = [8/(sinθ+ 4cosθ)]

Sketch the graph of r = 3sinθ, for 0 ≤ θ ≤ 2π

• Make a table for plot points

θ	r	(θ,r)
0	0	(0,0)
[(π)/4]	[(3√2 )/2]	([(π)/4],2.12)
[(π)/2]	3	([(π)/2],3)
[(3π)/4]	[(3√2 )/2]	([(3π)/4],2.12)
π	0	(π,0)
[(5π)/4]	− [(3√2 )/2]	([(5π)/4], − 2.12)
[(3π)/2]	− 3	([(3π)/2], − 3)
[(7π)/4]	− [(3√2 )/2]	([(7π)/4], − 2.12
2π	0	(2π,0)

Graph the function

Polar Coordinates

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• Intro 0:00
• Polar Coordinates 0:09
• Definition
• Example
• Converting Polar Coordinates 1:49
• Example: Convert Polar Equation to Cartesian Coordinates 3:06
• Example: Convert Polar Equation to Cartesian Coordinates 5:21
• Example: Find the Polar Representation 6:51
• Example: Find the Polar Representation 8:39
• Example: Sketch the Graph 10:02