Convert r = -2 sin θ to rectangular Step by Step

Step by step

The question

Question: write r = −2 sin θ in x and y.

r = −2 sin θ

1 Write the rules

The rules between polar and x, y: x = r cos θ, y = r sin θ, and r² = x² + y²: Pythagoras.

x = r cos θ, y = r sin θ, r² = x² + y²

2 Multiply both sides by r

y is r sin θ.

To get r sin θ, we multiply both sides by r.

r × r = −2r sin θ

Careful!

Not only the right side.

The left side is multiplied by r too.

Careful: not r = −2r sin θ ✗

On the left, r times r is r².

r² = −2r sin θ

3 Use the rules

r² is x² + y².

And r sin θ is y.

So x² + y² = −2y.

x² + y² = −2y

4 Complete the square

Move −2y to the left: it becomes + 2y.

Group the y terms.

x² + (y² + 2y) = 0

Half of 2 is 1.

Squared: 1.

Add 1 to both sides.

x² + (y² + 2y + 1) = 1

2 ÷ 2 = 1, 1 × 1 = 1

The bracket is a full square: (y + 1)².

So x² + (y + 1)² = 1.

x² + (y + 1)² = 1

5 Draw it

Draw the polar grid.

A circle: center (0, −1), radius 1.

6 Check

Let's check with one polar point: (−2, 90°).

Put θ = 90° in: r = −2 sin 90° = −2.

So the point is (0, −2).

Put it in.

It gives 1, the right side.

Right!

0² + (−2 + 1)² = 1

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.

Answer