Step by step
The question
Question: write the polar point (3, 180°) 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 180° from the right side of the x-axis, going up and round.
Then go out 3.
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: 3 times cos 180°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 180° is −1: the same size as cos 0°, from the special triangles.
It is negative: the point is left of the y-axis.
Put it in.
3 times −1 is −3.
Now y: 3 times sin 180°.
sin 180° is 0: the same size as sin 0°, from the special triangles.
It is positive here.
Put it in.
3 times 0 is 0.
So the point is (−3, 0).
3 Check
and straight to the y-axis.
The grid shows x −3 and y 0.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.