Pure 2.5 - Polar Coordinates

Question 1

What are polar coordinates?

Answer 1

An alternative method of describing the position of a point 𝑝 in a two-dimensional space
Two measurements are required:
The distance from the pole (usually the origin), 𝑟
And the angle measured anticlockwise from the initial line (usually the positive 𝑥-axis), θ
Polar coordinates are written as (𝑟,θ)

Question 2

How would you convert from cartesian form to polar form?

Answer 2

𝑥=𝑟cos(θ)
𝑦=𝑟sin(θ)
𝑟²=𝑥²+𝑦²
θ=arctan(𝑦/𝑥)

Question 3

Describe the polar graph 𝑟=𝑎.

Answer 3

𝑟=𝑎 is a circle with centre 𝑂 and radius 𝑎

Question 4

Describe the polar graph θ=α.

Answer 4

θ=α is a half-line through 𝑂 and making an angle α with the initial line

Question 5

Describe the polar graph r=𝑎θ.

Answer 5

r=𝑎θ is a spiral starting at 𝑂

Question 6

Describe how to find the area of sector bounded by a polar curve.

Answer 6

The area of a sector bounded by a polar curve and the half-lines θ=a and θ=b, where θ is in radians is given by the formula
A=½∫ₐᵇ𝑟²dθ

Question 7

How would you find a tangent of a polar curve that is parallel to the initial line?

Answer 7

Set d𝑦/dθ=0

Question 8

How would you find a tangent of a polar curve that is perpendicular to the initial line?

Answer 8

Set d𝑥/dθ=0