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
