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Lecture Comments (3)

1 answer

Last reply by: Him Tam
Tue Jul 30, 2013 6:24 PM

Post by Akshay Tiwary on February 23, 2013

Just a clarification. At about 10:00 when he sketches the graph, he mentions that the equation represents a "squiggly figure" while infact it does represent a circle centered at (1,0) with radius 1.

0 answers

Post by Cedrick Mah on April 10, 2012

at 2:46, its opposite/adjacent. Just a minor error, but great videos!!

Related Articles:

Polar Coordinates

  • Polar coordinates:
  • Used to express more complex, rounded shapes
  • Converting to Cartesian: ,
  • Converting to Polar: ,

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
0
(2π,0)
Graph the function

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polar Coordinates

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • 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