Step by step
The question
Question: graph r = 1 − 2 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.
So θ from −90° to 90° is enough: the rest is the mirror.
2 Make a table
A table: 7 angles, 30° apart, from -90° to 90°.
The angles go along the top.
sin of each angle, from the unit circle, to 2 decimal places: −1, −0.87, −0.5, and so on.
Then r = 1 minus 2 times that.
For -90°: 1 − 2 × (−1) is 3.
3 Plot the points
The polar grid: circles 1, 2, 3 around the center, and a line for every 30°.
θ = -90°: turn -90°, that is down, then go out 3.
θ = -60°: turn -60°, go out 2.7.
And so on.
At 60°, r is negative: −0.7.
We turn 60°, then go 0.7 the other way, backwards through the center.
Careful!
A negative r is not nothing.
We go backwards, through the center, to the other side.
4 Join and mirror
Join the points in order, with a smooth curve.
The other half is the mirror image in the y-axis.
5 Name it
1 is smaller than 2: r turns negative for some angles, and the curve makes a small loop inside.
So it is a limaçon with an inner loop.
6 Check
Let's check one angle we did not use: θ = 180°.
sin 180° is 0.
So r is 1: turn 180°, go out 1.
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 and mirror.
Then: name it.
Last: check.