Flashcards in Chapter 10 Review Deck (21):

1

## Polar Coordinates

### used to plot points on a sphere

2

## (r,θ)

###
r - distance from the pole

θ - angle measured counterclockwise from the polar axis

3

## How to name 4 polar representations

###
Take first coordinate

Add or subtract 2pi

Add or subtract pi from both the original and the new

4

## changing polar coordinates to cartesian coordinates

###
*plug in (r,θ) into each of these equations to find (x,y)

x = rcosθ

y = rsinθ

r^2 = x^2 + y^2

θ = inverse tan (y/x)

5

## Changing cartesian coordinates to polar coordinates

###
*find r by plugging into r^2 = x^2 + y^2 and θ by plugging into inverse tan

r^2 = x^2 + y^2

θ = inverse tan (y/x)

6

## Horizontal Line

###
y = b

r = b / sin θ

OR

r = b * cscθ

7

## Vertical Line

###
x = a

r = a / cosθ

oR

r = a * secθ

8

## Line Through the Pole

### θ = k

9

## Sin Coordinate values

###
θ sin θ

0 0

π/6 1/2

π/2 1

3π/2 -1

11π/6 -1/2

*symmetrical over y-axis

10

## Cos Coordinate Values

###
θ cos θ

0 1

π/3 1/2

π/2 0

2π/3 -1/2

π -1

*symmetrical over x-axis

11

## Circles

###
r = ± 2a cosθ

circle on the x-axis

through pole , radius = a

r = ± 2a sinθ

circle on the y-axis, through pole, radius = a

12

## Cardioid

###
r = a ± acosθ

on x axis

r = a ± asinθ

on y axis

13

## Limacon

###
r = a ± bcosθ

r = a ± bsinθ

If abs. value of....

a/b is greater than or equal to 2, then: convex limacon

1 < abs value a/b < 2: dimpled limacon

a/b < 1 then: limacon with inner loop

a/b = 1 : cardioid

14

## Rose

###
r = acos(nθ) r = asin(nθ)

# petals: n if n is odd, 2n if n is even

length of petal: a

first petal: set r = absvalue a and solve for θ

next petal: +2π/# petals

15

## Spiral of Archimedes

###
r = aθ (spiral out) r = a/θ (spiral in)

Use: θ±π

θ r

0

π/4

π/2

3π/4

π

5π/4

3π/2

7π/4

2π

16

## Polar Area

### one half the integral of theta 1 to theta 2 of r squared dθ

17

## Slope of a polar curve

###
dy/dx = (d/dθ)(rsinθ) / (d/dθ)(rcosθ)

18

## Tangent lines of Polar Curves

###
horizontal when dy/dθ = 0 and dx/dθ does not = 0

Vertical (vice versa)

If BOTH equal 0, use limits to determine it is is horizontal or vertical

19

## Steps to finding both horizontal and vertical tangent lines

###
1. derivative

2. set top equal to zero for horiz, set bottom equal to zero for vert.

3. if the same answer for both, use a limit to find out

4. plug each value into either y=rsinθ or x=rcosθ`

20

## Finding tangent lines at the pole

###
1. set function equal to zero

2. plug in values to tanθ

21