Core Pure - Polar Coordinates

Question 1

What do polar coordinates do?

Answer 1

Give us an alternate way to map points in a @D space

Question 2

What actually is a polar coordinate?

Answer 2

You find the distance from the pole (usually 0,0) and the angle made (usually with the x axis)

Question 3

What does a general polar coordinate grid look like?

Answer 3

Question 4

How do you convert between polar coordinates and cartesian coordinates?

Answer 4

You will often need to use trigonometric identities

Question 5

What form are polar coordinates usually given in?

Answer 5

Question 6

What are the 3 standard forms for polar coordinates?

Answer 6

Question 7

What is:

Answer 7

Question 8

What is:

Answer 8

Question 9

What is:

Answer 9

Question 10

What is the method to sketch polar coordinates?

Answer 10

Draw a table and test values
Usually multiplies of π/4 or π/3

Question 11

What do you know about these 2 forms of curves?
r=cos/sin(nx)

Answer 11

Question 12

What are the 2 new shape called?

Answer 12

A cardioid (dimple)
Egg shape

Question 13

Draw a cardioid

Answer 13

Question 14

What curves do you need to know in more detail?

Give the form and 2 fact about them?

Answer 14

They can all be divided into 2 catagories

Question 15

What are the 2 categories?
(Of advanced polar curves)

Answer 15

Egg shape
Dimple

Question 16

Draw an “egg shape” and state when it occurs

Answer 16

Question 17

Draw a cardioid and state when it occurs

Answer 17

Question 18

What is the formular for areas of polar coordinates?

Answer 18

Question 19

What is a really useful thing to remember when working out areas with polar coordinates?

Answer 19

There is no x and y axis so integrating below π will still work (and return a positive answer)

If a shape is symmetrical you can integrate to find the area of a quarter of it then multiply by 4 etc

You will often be pulling out a squared terms to the front of the integral

Question 20

What do you need to remember can’t be done when integrating with polar coordinates?
Why?

Answer 20

For a shape with n loops you need to find the area of the whole shape and not just 1 loop

This is because the r makes it so that it includes the r≤0 as well

Question 21

How do you find tangents to polar curves?

Answer 21

Using derivatives

Question 22

Where will there usually a tangent that you ignore?

Answer 22

The center

Question 23

Where are the tangents?

Answer 23

Parallel or perpendicular to the initial line
Eg:

Question 24

How do you work out a tangent when it is parallel to the initial line?

Answer 24

Question 25

How do you find the tangent when it is perpendicular to the initial line?

Answer 25

Question 26

What is the general method for working out these derivatives?

Answer 26

You work out which one you need x or y then use eg: x=rcosθ
Then rearrange for r and sub it into the polar curve
Solve it and then put your θ value into the polar equation for the distance

Question 27

What is a very common mistake in this whole topic especially big mark questions?
(Polar)

Answer 27

The distance may be in terms of a. Since you set it equal to 0 they go but get forgotten in the actual answer