Convert θ = 45° to rectangular

Step by step

The question

Question: write θ = 45° 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 Use tan θ = y/x

y/x is r sin θ over r cos θ.

The r's cancel: it is tan θ.

So y/x = tan 45°.

tan 45° is 1, from the special triangles.

Put it in.

3 Get y alone

Multiply both sides by x: y = x.

Careful!

The angle is not the slope.

The slope is tan 45°.

4 Draw it

Draw the polar grid.

A straight line through the center, at 45°.

5 Check

Let's check with one polar point: (2, 45°).

It is on the line θ = 45°.

Its y divided by its x is the same as tan 45°.

Right!

Let's look back.

First: write the rules.

Then: use tan θ = y/x.

Then: get y alone.

Then: draw it.

Last: check.