Polar Coordinates Flashcards Preview

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Flashcards in Polar Coordinates Deck (25)
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1

Polar form

(r,θ) where r is the modulus and θ the anti-clockwise angle in radians from the positive x-axis

2

Graph of r=aθ

A spiral about the origin, growing 2aπ wider each spiral

3

Cartesian and polar conversion values

x^2 + y^2 = r^2
x = r cosθ
y = r sinθ
Use trigonometric manipulation

4

Graph of r = a

Gives a circle with radius a

5

Graph of θ = a

Half-line from the origin at angle a

6

Sketching from table of values:

Find r for θ = 0, π/2, π, 3π/2 and 2π or those values divided by a if you have aθ
Remove negative r
Sketch with the correct shape
Repeat where appropriate from 0 to 2π

7

r = a(p+qcosθ) where p = |q|

A cardioid, almost heart shaped but it circles rather than having a pointy end

8

r = a(p+qcosθ) where p >= q and p>|2q|

An oval or egg shape

9

r = a(p+qcosθ) where p >= q and |q| < p < |2q|

A dimple, cardioid shape but the centre of the dimpled section is not at the origin

10

Area under a polar curve formula

1/2 ∫ r^2 dθ between angles α and β

11

cos^2 x simplified

1/2 + 1/2 cos(2x)

12

sin^2 x simplified

1/2 - 1/2 cos(2x)

13

Integrating from 0 to 2π

2 π
1. 1/2 (constant)^2 ∫ r^2 dθ
0
2. Expand r^2
3. Replace cos^2 x and sin^2 x
4. Integrate each part and sub in the numbers

14

One loop of a polar rose

Take the first two θ values that give r = 0 and integrate between those

15

dy/dx when x = cos(t) and y = sin(t)

(dy/dt) / (dx/dt) = -cot(t)

16

Parallel to the initial line then

dy/dθ = 0

17

Perpendicular to the initial line then

dx/dθ = 0

18

Points parallel or perpendicular to the initial line

Start with y = r sinθ or x = r cosθ depending on which will be set to 0
Substitute the polar form for r
Differentiate and set to 0
Solve for θ
Find r for each θ and put into polar form

19

tanθ to cosθ and sinθ

Create a right angled triangle

20

Finding a tangent or normal

Put the y or x into cartesian using y = rsinθ or x = rcosθ and substitute that with a generic r and cosθ or sinθ

21

Angles below the x axis

Clockwise from π to 2π

22

Graph of r = acos(theta)

Circle radius a/2 about the point where x = a/2

23

r = sec or cosec with addition formulae

Put rcos/rsin= and expand

24

Graph of r = asin(theta)

Circle radius a/2 about the point where y = a/2

25

Area between polar curves

Subtract the square of each equation and do one integration