Step by step
The question
Question: write the polar point (−2, 45°) in rectangular form (x, y).
1 Plot the point
Draw the polar grid: circles 1, 2, 3 around the center, and lines for the angles.
Turn 45° from the right side of the x-axis, going up and round.
r is negative.
So we face 45°,
then go 2 backwards, the other way.
2 Use x = r cos θ, y = r sin θ
For any polar point: x = r cos θ and y = r sin θ.
They come from the right triangle under the point.
First x: −2 times cos 45°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 45° is √2/2.
We know it from the special triangles.
Put it in.
−2 times √2/2 is −√2.
Now y: −2 times sin 45°.
sin 45° is √2/2.
We know it from the special triangles.
Put it in.
−2 times √2/2 is −√2.
So the point is (−√2, −√2).
3 Check
Let's check on the grid.
From the point, go straight to the x-axis
and straight to the y-axis.
The grid shows x about −1.4 and y about −1.4.
−√2 is about −1.4.
−√2 is about −1.4.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.