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