Step by step
The question
Question: graph r = 4 sin θ.
θ is the angle we turn, r is how far we go out.
1 Check the symmetry
sin of 180° minus θ is the same as sin θ.
So the graph is its own mirror image in the y-axis.
For this circle, θ from 0° to 180° already draws all of it.
2 Make a table
A table: 7 angles, 30° apart, from 0° to 180°.
The angles go along the top.
sin of each angle, from the unit circle, to 2 decimal places: 0, 0.5, 0.87, and so on.
Then r = 4 times that.
For 0°: 4 × 0 is 0.
3 Plot the points
The polar grid: circles 1, 2, 3 around the center, and a line for every 30°.
θ = 0°: turn 0°, then go out 0.
θ = 30°: turn 30°, go out 2.
And so on.
Careful!
The point is not x = θ, y = r.
We turn θ, then walk out r along that line.
4 Join the points
Join the points in order, with a smooth curve.
5 Name it
r = 4 sin θ is a circle through the center, with diameter 4.
So it is a circle.
6 Check
Let's check one angle we did not use: θ = 45°.
sin 45° is 0.71.
So r is 2.8: turn 45°, go out 2.8.
The point is on our curve.
Right!
Let's look back.
First: check the symmetry.
Then: make a table.
Then: plot the points.
Then: join the points.
Then: name it.
Last: check.