Polar Coordinates Flashcards
Polar form
(r,θ) where r is the modulus and θ the anti-clockwise angle in radians from the positive x-axis
Graph of r=aθ
A spiral about the origin, growing 2aπ wider each spiral
Cartesian and polar conversion values
x^2 + y^2 = r^2
x = r cosθ
y = r sinθ
Use trigonometric manipulation
Graph of r = a
Gives a circle with radius a
Graph of θ = a
Half-line from the origin at angle a
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
r = a(p+qcosθ) where p = |q|
A cardioid, almost heart shaped but it circles rather than having a pointy end
r = a(p+qcosθ) where p >= q and p>|2q|
An oval or egg shape
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
Area under a polar curve formula
1/2 ∫ r^2 dθ between angles α and β
cos^2 x simplified
1/2 + 1/2 cos(2x)
sin^2 x simplified
1/2 - 1/2 cos(2x)
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 numbersOne loop of a polar rose
Take the first two θ values that give r = 0 and integrate between those
dy/dx when x = cos(t) and y = sin(t)
(dy/dt) / (dx/dt) = -cot(t)
Parallel to the initial line then
dy/dθ = 0
Perpendicular to the initial line then
dx/dθ = 0
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
tanθ to cosθ and sinθ
Create a right angled triangle
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
Angles below the x axis
Anti-clockwise from π to 2π
Graph of r = acos(theta)
Circle radius a/2 about the point where x = a/2
r = sec or cosec with addition formulae
Put rcos/rsin= and expand
Graph of r = asin(theta)
Circle radius a/2 about the point where y = a/2
Area between polar curves
Subtract the square of each equation and do one integration
