Polar Coordinates
What is sin(-θ) equal to?
-sin(θ)
How to convert from rectangular equation to polar equation
1. Plug r into rcosθ and rsinθ for x and y respectively 2. Factor out r 3. Divide by what r is being multiplied to
How to convert from rectangular coordinates to polar coordinates
1. Plug x and y into r=√x²+y² and θ=tan⁻¹(y/x) 2. Solve and try to put into exact form 3. Check original coordinates to make sure the polar coordinates is in the same quadrant
How many petals are there when you graph cos(nθ)?
2n number of petals
If the graph is rotated about the pole by α, change equation to...
Conic: r=(ep)/(1-ecos(θ-α)) Circle: r=kcos(θ-α) or r=ksin(θ-α) Line: r=ksec(θ-α) or r=csc(θ-α)
True or false? If an equation doesn't pass the symmetry tests, then it is not symmetrical.
FALSE
How to convert from polar equation to rectangular equation
If given what r equals 1. Plug r into r=√x²+y² 2. Square r If given what θ equals 1. Plug θ into θ=tan⁻¹(y/x) 2. tanθ=y/x and solve tanθ 3. Multiply x over
How can you change a polar coordinate to be in the same quadrant as the rectangular coordinate
Make r negative OR add 180° or π to θ
If z=ι, what is a and b?
a=0 and b=1
If z=any real number, what is a and b?
a=the real number and b=0 ex: z=1 a=1 b=0
In r=(ep)/(1-cosθ) how does the +/- affect the graph?
above or below the major axis right or left of the major axis
General Equation for a Circle (particularly if it doesn't touch the pole)
a²=r²+k²-2rkcos(θ-α)
In a limaçon: When it's -bsinθ, the graph is
below the x axis
In r=a+bcosθ and r=a-bcosθ, what limaçons is |a/b|=1?
cardioid (touches the origin)
In r=a+bcosθ and r=a-bcosθ, what limaçons is it when a=0 OR b=0?
circle
When the angle is negative, which direction do you go?
clockwise
What is cos(-θ) equal to?
cos(θ)
In r=(ep)/(1-cosθ) how does the cosθ/sinθ affect the graph?
cosθ → along the x axis sinθ → along the y axis
When the angle is positive, which direction do you go?
counterclockwise
In r=a+bcosθ and r=a-bcosθ, what limaçons is 1<|a/b|<2?
dimple
What are the multiple representations of polar points?
ex: (3, 45°) 1. -r and θ+180° (-3, 225°) 2. -r and θ−180° (-3, -135°) 3. r and θ+360°n (3, 405°) (3, 765°)
How to find the roots when z just equals any real number or ι
ex: Cube roots of 8 1. Convert to polar form of z form ex: z=8 a=8 b=0 r=√8²+0²=8 cosθ=8/8=1 sinθ=0/8=0 θ=0 z=8cis0 2. Plug polar form of z form into roots formula ³√8cis0=³√8cis((0+360°n)/3)=2cis(0+120°n) 3. n number of answers and convert them to a+bι form ex: 2cis0=2(cos0+ιsin0)=2 2cis120=2(cos120+ιsin120)=-2+1.73ι 2cis240=2(cos240+ιsin240)=-2-1.73ι
How to solve systems of polar equations #1 Basic System
ex: r=1 r=2cosθ 1. Identify what shape the equations make and graph them to see how many intersections there are ex: r=1 → circle r=2cosθ → circle passes through pole 2 points of intersection 2. Set r's equal to each other and solve to find θ ex: 1=2cosθ 1/2=cosθ cos⁻¹(1/2)=θ θ=60°, 120° 3. Plug θ back into one of the equations to find r ex: Don't have to because already know r=1 Answers: (1, 60°), (1, 120°)
How to sketch a graph of a conic section when given the general equation with focus at the pole
ex: r=12/(3-6cosθ) 1. Factor out number if the denominator isn't 1 ex: 12/3(1-2cosθ) 2. Simplify numerator and denominator if possible ex: 4/(1-2cosθ) 3. Use the eccentricity to identify what conic it is ex: e=2 → hyperbola 4. Use the +/- and cos/sin of the denominator to identify the rest of the shape ex: -cosθ → hyperbola is left of the y axis with the right part of the hyperbola opening to the pole
How to solve systems of polar equations #4 The Unit Circle
ex: r=sin3θ r=cos3θ 1. Identify what shape the equations make and graph them to see how many intersections there are ex: r=sin3θ → petals r=cos3θ → petals 4 intersections and 1 of the intersections is (0, 0) 2. Set the r's equal to each other to solve for θ ex: sin3θ=cos3θ sin3θ-cos3θ=0 cos3θ(tan3θ-1)=0 cos3θ=0 cos⁻¹(0)=3θ 90°+360°n=3θ → θ=30°+120°n 270°+360°n=3θ → θ=90°+120°n tan3θ-1=0 tan3θ=1 tan⁻¹(1)=3θ 45°+180°n=3θ → θ=15°+60°n 3. Plug all possible θ answers between 0 and 360 back into both equations to find r ex: θ: 30 90 150 210 270 330 r=sin3θ: 1 -1 1 -1 1 -1 r=cos3θ: 0 0 0 0 0 0 *r's DON'T match so these answers aren't real θ: 15 75 135 195 255 315 r=sin3θ: √2/2 -√2/2 √2/2 -√2/2 √2/2 -√2/2 r=cos3θ: √2/2 -√2/2 √2/2 -√2/2 √2/2 -√2/2 *r's DO match so these answers are real Answers: (0, 0), (√2/2, 15°), (-√2/2, 75°), (√2/2, 135°)
How to convert from polar form of z form back to z form
ex: z=2cis330° 1. Expand it using cisθ=(cosθ+ιsinθ) ex: z=2(cos330°+ιsin330°) 2. Simplify ex: a=2cos330° b=2ιsin330° z=√3-ι
In r=a+bcosθ and r=a-bcosθ, what limaçons is |a/b|<1?
inner loop (touches the origin) *when graphing this the inside circle's endpoint is |a-b| and outside circle's end point is a+b
In a limaçon: When it's -bcosθ, the graph is
left of the y axis
For polar equations of conics and lines: What does a hyperbola look like when it's -cosθ?
left of the y axis and the right part of the hyperbola opens facing the pole
What is the shape of this equation? r=ksecθ or r=kcscθ
line perpendicular to polar axis k=distance from polar axis to the line
For polar equations of conics and lines: What does an ellipse look like when it's -sinθ?
most of the ellipse is above the x axis and the pole is left of the center
For polar equations of conics and lines: What does an ellipse look like when it's cosθ?
most of the ellipse is on the left side of the y axis and the pole is right of the center
For polar equations of conics and lines: What does an ellipse look like when it's -cosθ?
most of the ellipse is on the right side of the y axis and the pole is left of the center
For polar equations of conics and lines: What does an ellipse look like when it's sinθ?
most of the ellipse is under the x axis and the pole is right of the center
How many petals are there when you graph sin(nθ)?
n number of petals
In r=a+bcosθ and r=a-bcosθ, what limaçons is |a/b|≥2?
no dimple, no inner loop (in the image the left side is straight)
General Equation of Conics with Focus at the Pole
r=(ep)/(1-ecosθ) p=distance from the focus to the directrix e=eccentricity
Standard equation for polar coordinates
r=a+bcosθ r=a+bsinθ
θ=π/2 Test for Symmetry
replace (r, θ) with (-r, -θ) If the same equation after test, passes and points are reflected over the y axis
Pole Test for Symmetry
replace r with -r If the same equation after test, passes and points are reflected over the origin
For polar equations of conics and lines: What does a hyperbola look like when it's cosθ?
right of the y axis and the left part of the hyperbola opens facing the pole
What happens when a gets smaller?
the inner circle gets bigger until it reaches 0 as a circle, then becomes the outer circle and then becomes a cardioid, then dimple, then no dimple/no inner loop
What happens when a gets bigger?
the inner circle gets smaller and becomes a cardioid, then dimple, then no dimple/no inner loop
For polar equations of conics and lines: What does a hyperbola look like when it's -sinθ?
under the x axis and the right part of the hyperbola opens facing the pole
Standard equation for rectangular coordinates
y=mx+b
How to convert from polar coordinates to rectangular coordinates
1. Plug r and θ into x=rcosθ and y=rsinθ 2. Solve
In a limaçon: When it's bsinθ, the graph is
above the x axis
For polar equations of conics and lines: What does a hyperbola look like when it's sinθ?
above the x axis and the left part of the hyperbola opens facing the pole
What is the shape of this equation? r=kcosθ or r=ksinθ
circle passing through the pole k=diameter
In r=(ep)/(1-cosθ) how does the eccentricity affect the graph?
e<1 → ellipse e=1 → parabola e>1 → hyperbola
How to find the polar equation from a conic's description
ex: Ellipse: Focus at the pole, corresponding directrix parallel to the polar axis, 4 units below the pole, eccentricity 2/3. 1. Draw graph (optional) 2. Dissect information given into r=ep/1-cosθ or r=ep/1-sinθ ex: "corresponding directrix parallel to the polar axis" → sinθ "4 units below the pole" → p=4 and sinθ is positive 3. Plug in values and simplify ex: r=(2/3*4)/(1-(2/3)sinθ) r=(8/3)/(1-(2/3)sinθ) (optional) r=8/(3-2sinθ)
How to sketch circle rotated about the pole by α
ex: r=-8cos(θ-70)° 1. Graph (k, α) to find the other end of the circle ex: (-8, 70°) 2. Find center by using α and the radius (d/2) ex: (-4, 70°)
How to solve systems of polar equations #5 Solve, then solve again but replace (r, θ) with (-r, θ+180°)
ex: r=2+2cosθ r=1+3sinθ 1. Identify what shape the equations make and graph them to see how many intersections there are ex: r=2+2cosθ r=1+3sinθ 3 intersections and 1 of the intersections is (0, 0) 2. Set r's equal to each other and solve for θ using Acosx+Bsinx=Ccos(x-D); C=√A²+B²; cosD=A/C; sinD=B/C ex: 2+2cosθ=1+3sinθ 2cosθ-3sinθ=-1 C=√2²+(-3)²=√13 cosD=2/√13 → D=56.31° sinD=-3/√13 → D= -56.31° *make sure the D corresponds to quadrant* so since cos is + and sin is -, then it's in the 4th quadrant so D=-56.31° √13cos(θ+56.31°)=-1 cos⁻¹(-1/√13)=θ+56.31° θ+56.31°=±106.1°+360°n θ=-56.31°±106.1°+360°n θ=49.8°, 197.6° 3. Plug θ back into 1 of the equations to find r ex: 2+2cos49.8°=3.29 2+2cos197.6=0.09 4. Check for missing solutions by replace one of the equations with (-r, θ+180°) ex: -r=2+2cos(θ+180°) *adding 180 just makes cos negative -r=2-2cosθ r=-2+2cosθ 5. Set r's equal to each other -2+2cosθ=1+3sinθ 2cosθ-3sinθ=3 √13cos(θ+56.31°)=3 θ=-56.31°±33.7°+360°n θ=270°, 337.9° (have to add 360° to fit range) r=-2, -0.15 Answers: (0, 0), (3.29, 49,8°), (0.09, 197.6°), (-2, 270°), (-0.15, 337.9°)
How to solve systems of polar equations #3 Convert to rectangular, solve it there, then convert back to polar
ex: r=2cosθ r=1+cosθ 1. Convert equations to rectangular by using x=rcosθ → cosθ=x/r and r=√x²+y² ex: r=2cosθ r=2x/r r²=2x x²+y²=2x √x²+y²=√2x r=1+cosθ r=1+(x/r) r²=r+x x²+y²=(√x²+y²)+x 2. Substitute to solve for x ex: 2x=(√2x)+x x=√2x x²=2x x²-2x=0 x(x-2)=0 x=0, 2 3. Plug x back into one of the rectangular equations to find y ex: 0²+y²=2(0) y=0 2²+y²=2(2) 4+y²=4 y=0 Rectangular answer: (0, 0), (2, 0) 4. Convert answers to polar r=√0²+0²=0 θ=arctan(0/0) θ=0 r=√2²+0²=2 θ=arctan(0/2) θ=0 Polar Answers: (0, 0), (2, 0)
How to solve systems of polar equations #2 Use a graph
ex: r=cosθ r=1-cosθ 1. Identify what shape the equations make and graph them to see how many intersections there are ex: r=cosθ → circle through pole r=1-cosθ → cardioid 3 points of intersection and 1 of the intersections is (0, 0) 2. Set r's equal to each other and solve to find θ ex: cosθ=1-cosθ 2cosθ=1 cosθ=1/2 cos⁻¹(1/2)=θ θ=60°, 120° 3. Plug θ back into one of the equations to find r ex: r=cos(60) → r=1/2 r=cos(120) → r=1/2 Answer: (0, 0), (1/2, 60°), (1/2, 120°)
How to convert from z form to polar form of z form
ex: z=√3-ι 1. Find r and θ using r=√a²+b², cosθ=a/r, and sinθ=b/r ex: a=√3 b=-1 r=√(√3)²+(-1)²=2 cosθ=√3/2 sinθ=-1/2 θ=330° *make sure angle corresponds with quadrant* so since cos is positive and sin is negative, it would be in quadrant 4 2. Plug into z=rcisθ ex: z=2cis330°
Polar Axis Test for Symmetry
replace θ with -θ If the same equation after test, passes and the points are reflected over the x axis
In a limaçon: When it's bcosθ, the graph is
right of the y axis
