Step by step
The question
Question: write the polar point (6, 210°) 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 210° from the right side of the x-axis, going up and round.
Then go out 6.
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: 6 times cos 210°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 210° is −√3/2: the same size as cos 30°, from the special triangles.
It is negative: the point is left of the y-axis.
Put it in.
6 times −√3/2 is −3√3.
Now y: 6 times sin 210°.
sin 210° is −1/2: the same size as sin 30°, from the special triangles.
It is negative: the point is below the x-axis.
Put it in.
6 times −1/2 is −3.
So the point is (−3√3, −3).
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 −5.2 and y about −3.
−3√3 is about −5.2.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.