Step by step
The question
Question: write the polar point (3, 2π over 3) 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 2π over 3 from the right side of the x-axis, going up and round.
Then go out 3.
π is 180°.
So 2π over 3 is 120°.
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 120°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 120° is −1/2: the same size as cos 60°, from the special triangles.
It is negative: the point is left of the y-axis.
Put it in.
3 times −1/2 is −3/2.
Now y: 3 times sin 120°.
sin 120° is √3/2: the same size as sin 60°, from the special triangles.
It is positive here.
Put it in.
3 times √3/2 is 3√3/2.
So the point is (−3/2, 3√3/2).
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 −1.5 and y about 2.6.
3√3/2 is about 2.6.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.