Step by step
The question
Question: graph r = 2 cos θ.
θ is the angle we turn, r is how far we go out.
1 Check the symmetry
cos of minus θ is the same as cos θ.
So turning up or down by the same angle gives the same r: the graph is its own mirror image in the x-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.
cos of each angle, from the unit circle, to 2 decimal places: 1, 0.87, 0.5, and so on.
Then r = 2 times that.
For 0°: 2 × 1 is 2.
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 2.
θ = 30°: turn 30°, go out 1.7.
And so on.
At 120°, r is negative: −1.
We turn 120°, then go 1 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 the points
Join the points in order, with a smooth curve.
5 Name it
r = 2 cos θ is a circle through the center, with diameter 2.
So it is a circle.
6 Check
Let's check one angle we did not use: θ = 45°.
cos 45° is 0.71.
So r is 1.4: turn 45°, go out 1.4.
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.