Polar Coordinates

Question 1

Polar form

Answer 1

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

Question 2

Graph of r=aθ

Answer 2

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

Question 3

Cartesian and polar conversion values

Answer 3

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

Question 4

Graph of r = a

Answer 4

Gives a circle with radius a

Question 5

Graph of θ = a

Answer 5

Half-line from the origin at angle a

Question 6

Sketching from table of values:

Answer 6

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

Question 7

r = a(p+qcosθ) where p = |q|

Answer 7

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

Question 8

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

Answer 8

An oval or egg shape

Question 9

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

Answer 9

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

Question 10

Area under a polar curve formula

Answer 10

1/2 ∫ r^2 dθ between angles α and β

Question 11

cos^2 x simplified

Answer 11

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

Question 12

sin^2 x simplified

Answer 12

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

Question 13

Integrating from 0 to 2π

Answer 13

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

Question 14

One loop of a polar rose

Answer 14

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

Question 15

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

Answer 15

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

Question 16

Parallel to the initial line then

Answer 16

dy/dθ = 0

Question 17

Perpendicular to the initial line then

Answer 17

dx/dθ = 0

Question 18

Points parallel or perpendicular to the initial line

Answer 18

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

Question 19

tanθ to cosθ and sinθ

Answer 19

Create a right angled triangle

Question 20

Finding a tangent or normal

Answer 20

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

Question 21

Angles below the x axis

Answer 21

Anti-clockwise from π to 2π

Question 22

Graph of r = acos(theta)

Answer 22

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

Question 23

r = sec or cosec with addition formulae

Answer 23

Put rcos/rsin= and expand

Question 24

Graph of r = asin(theta)

Answer 24

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

Question 25

Area between polar curves

Answer 25

Subtract the square of each equation and do one integration