2 - 9 - polar coordinates

Question 1

polar coordinates

Answer 1

(r, θ)
theta is from 0-2pi and is measured anticlockwise

Question 2

r = θ graph

Answer 2

spiral

Question 3

r = a sin pθ graph

Answer 3

graph with p petals starting at (0,0)

Question 4

r = a cos pθ graph

Answer 4

graph with p petals staring at (a,0)

Question 5

min/max of r

Answer 5

dr/dθ = 0

Question 6

tangents at the pole

Answer 6

if r = f(θ)
the line θ = a is a tangent if f(a) = 0 but f(a) >0 on one side of the line

Question 7

changing from polar to Cartesian - coordinates

Answer 7

(r,θ)
(r cosθ, r sinθ)

Question 8

changing from polar to Cartesian - equations

Answer 8

r = √x^2 + y^2
tanθ = y/x
x = r cosθ
y = r sinθ

Question 9

area enclosed by a polar curve

Answer 9

A = 1/2 ∫ r² dθ

Question 10

are between 2 curves

Answer 10

find intersection points
then calc part of area bounded by each curve

Question 11

r = a(1+cos) graph

Answer 11

cartoid with point on the left

Question 12

r = p + qcos graph

Answer 12

if p>=2q no dimple - egg
if p< 2q then a dimple
if p = q then point

Question 13

r = a graph

Answer 13

circle radius a

Question 14

θ = ⍺ graph

Answer 14

half line of an angle ⍺

Question 15

r = a(1+sin) graph

Answer 15

cartoid but with the point at the top

Question 16

tangents - petals

Answer 16

look at what angles the petals are at - will be those half lines

Question 17

tangents - eggs/ cartoids

Answer 17

x = rcosθ
y = r sinθ
to initial line:
when perp dx/dθ = 0
when parallel dy/dθ = 0