Step by step
The question
Question: write r = 6 cos θ in x and y.
1 Write the rules
The rules between polar and x, y: x = r cos θ, y = r sin θ, and r² = x² + y²: Pythagoras.
2 Multiply both sides by r
x is r cos θ.
To get r cos θ, we multiply both sides by r.
Careful!
Not only the right side.
The left side is multiplied by r too.
On the left, r times r is r².
3 Use the rules
r² is x² + y².
And r cos θ is x.
So x² + y² = 6x.
4 Complete the square
Move 6x to the left: it becomes − 6x.
Group the x terms.
Half of −6 is −3.
Squared: 9.
Add 9 to both sides.
The bracket is a full square: (x − 3)².
So (x − 3)² + y² = 9.
5 Draw it
Draw the polar grid.
A circle: center (3, 0), radius 3.
6 Check
Let's check with one polar point: (6, 0°).
Put θ = 0° in: r = 6 cos 0° = 6.
So the point is (6, 0).
Put it in.
It gives 9, the right side.
Right!
Let's look back.
First: write the rules.
Then: multiply both sides by r.
Then: use the rules.
Then: complete the square.
Then: draw it.
Last: check.