Y2, C5- Polar Coordinates Flashcards
What does a polar coordinate equation look like
P(r, θ)
Where r = length from pole
θ = angle anticlockwise from x axis measured in radians
What is the Cartesian equation for r = 5
x^2 + y^2 = 25
What equations do you need to know for x and y when converting polar coordinates into Cartesian equations
r^2 = x^2 + y^2
x = r * cos θ
y = r * sin θ
θ = arctan(y/x)
Write r^2 = sin(θ + pi/4) in Cartesian form
r^2 = sinθcos(pi/4) + cosθsin(pi/4)
r^2 = (1/root(2))sinθ + (1/root(2))cosθ
root(2)r^3 = rsinθ + rcosθ
root(2)(x^2 + y^2)^3/2 = x + y
What do polar equations usually start with (LHS)
r =
OR
r^2 =
Convert x^2 - y^2 = 5 into a polar equation
r^2 * cos^2(θ) - r^2 * sin^2(θ) = 5
r^2 (cos^2(θ) - sin^2(θ)) = 5
r^2 (cos(2θ)) = 5
r^2 = 5sec(2θ)
How would you sketch:
r = a
θ = alpha
r = aθ
1) Circle, radius a
2) Half line, angle alpha from +ve x axis
3) Spiral
What does the edexcel spec remove in polar coordinates
Negative values for r
How many +ve petals are there for the a polar equation in the form r^2 = a^2 * cos(3θ)
3 +ve petals
Coefficient before θ is the number of petals
r = a(p + qcosθ), what shape do you get if p = q
You get a cardioid (egg with dimple) where the curve reaches the origin when θ = pi
r = a(p + qcosθ), what shape do you get if p >= 2q
Egg / oval shape (if q = 0, circle centred at origin)
r = a(p + qcosθ), what shape do you get if q < p < 2q
Dimple shape, unlike cardioid, centre will NEVER be at origin as r > 0 and not equal to 0
Describe the shape of r = 2acosθ
Radius a, centre (a, 0)
Describe the shape of r = a(1 - cosθ)
Backwards dimple
Describe the shape of r = asec(θ)
Vertical straight line, x = a
What is the formula for the area of each sector of a polar petal
0.5 * r^2 * dθ
What is the integration formula polar integration
0.5 * int(r^2) dθ from beta to alpha
How do you find the strategy of combines shaded areas
Draw a line from origin to the intersection to split the shaded area and then integrate under each curve separately
When can you find tangents and normals to parametric curves
When they are parallel or perpendicular to the initial line
dx / dθ = 0
OR
dy / dθ = 0
With polar coordinates, what is (dy/dθ) / (dx/dθ) equal to?
dy / dx
When perpendicular to the initial line, what is dx/dθ equal to
0
When parallel to the initial line, what is dy/dθ equal to
0
Curve C has polar equation: r = 1 + 2cosθ, 0 < θ < pi/2
At point P on C, tangent to C is parallel to initial line. Find exact length of OP
y = rsinθ
y = 2sinθcosθ + sinθ
y = sin2θ + sinθ
dy / dθ = 4cos^(θ) + cosθ - 2 = 0
θ = (-1+root(33)) / 8
1 = 2cosθ = (3+root(33) / 4
What is the proof for r = p + qcosθ has a dimple if p < 2q
There will be 3 tangents rather than 2 (which an egg has)
Find the 3 tangents
sinθ = 0
OR
cosθ = -p / 2q
If p > 2q, cosθ < -1, no solutions so not other tangent
If p = 2q, cosθ = -1 which gives θ = pi which is already a solution and so not another tangent
