Polar Coordinates

Question 1

What are the equations relating polar and cartesian that are used too convert between points?

Answer 1

rcosθ = x
rsinθ = y
r² = x²+y²
θ = arctan(y/x)

Note always measure from the positive x axis therefore you must draw a graph

Question 2

What shape is represented by polar equation r=n ?

Answer 2

A circle centre the origin of radius n

Question 3

What shape is represented by the polar equation θ=nπ/k ?

Answer 3

A half line with argument nπ/k starting from the positive x-axis?

Question 4

What shape is represented by the polar equation r=aθ?

Answer 4

A spiral extending out anti-clockwise from the origin

Question 5

How do you sketch more complex polar curves such as r=asin(3θ) ?

Answer 5

• Sketch the trig function your curve relates too i.e sin(3θ) against r
• From your graph draw your θ agains r table (only where r≥0)
• Draw on your points and connect them with curved lines noting the angle at which the line leaves and enters the axis

Question 6

What are the three possible cases for polar curves with equation r=a(p+qcosθ) ?

Answer 6

p=q you draw a cusp (sharp point inwards)

q < p < 2q you draw a dimple (dull point inwards)

p ≥ 2q you get an egg shape (no point just stretched at one part)

Note if it is sin not cos the shape remains the same but the ‘interesting’ part is below the axis not to the left

Question 7

Whata are the steps for drawing a polar curve that has the form r=a(p+qcos(θ)) ?

Answer 7

• Figure out wether it is a cusp, dimple or egg
• Figure out whether the ‘interesting’ part happens to the left of the y-axis or beneath
• Draw on your points and connect them with curves (using your r-theta table and your trig graph)

Question 8

What is the formula for area of a polar curve that is given in the formula book?

Answer 8

1/2 ∫^β_α r² dθ

Note β and α must be in radians

Question 9

What is important to do when integrating polar curves?

Answer 9

• Draw your θ against r table where r≥0
• Draw the coordinates of known points on your graph using this table
• Find the origin (generally using known points or subbing values into your equation for r)

Question 10

What do you do if asked to find the points where the tangent to a polar curve is parallel to the intitial line?

Answer 10

• State y = rsin(θ) then sub in for r
• Find dy/dθ
• Set dy/dθ = 0
• Solve

Question 11

What do you do if asked to find the points where the tangent to a polar curve is perpendicular to the intitial line?

Answer 11

• State x = rcos(θ) then sub in for r
• Find dx/dθ
• Set dx/dθ = 0
• Solve