Step by step
The question
Question: write the polar point (4, 60°) 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 60° from the right side of the x-axis, going up and round.
Then go out 4.
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: 4 times cos 60°.
Careful!
x goes with cos, y goes with sin.
Swapped, we would get the point mirrored.
cos 60° is 1/2.
We know it from the special triangles.
Put it in.
4 times 1/2 is 2.
Now y: 4 times sin 60°.
sin 60° is √3/2.
We know it from the special triangles.
Put it in.
4 times √3/2 is 2√3.
So the point is (2, 2√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 2 and y about 3.5.
2√3 is about 3.5.
The same.
Right!
Let's look back.
First: plot the point.
Then: use x = r cos θ, y = r sin θ.
Last: check.