Step by step
The question
Question: write r = 4 sin 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
y is r sin θ.
To get r sin 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 sin θ is y.
So x² + y² = 4y.
4 Complete the square
Move 4y to the left: it becomes − 4y.
Group the y terms.
Half of −4 is −2.
Squared: 4.
Add 4 to both sides.
The bracket is a full square: (y − 2)².
So x² + (y − 2)² = 4.
5 Draw it
Draw the polar grid.
A circle: center (0, 2), radius 2.
6 Check
Let's check with one polar point: (4, 90°).
Put θ = 90° in: r = 4 sin 90° = 4.
So the point is (0, 4).
Put it in.
It gives 4, 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.