Worked examples
The question
Question: graph r = 4 cos θ.
θ is the angle we turn, r is how far we go out.
r = 4 cos θ
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.
cos(−θ) = cos θ: the same r above and below the x-axis
2 Make a table
A table: 7 angles, 30° apart, from 0° to 180°.
The angles go along the top.
0°
30°
60°
90°
120°
150°
180°
cos of each angle, from the unit circle, to 2 decimal places: 1, 0.87, 0.5, and so on.
1
0.87
0.5
0
−0.5
−0.87
−1
Then r = 4 times that.
For 0°: 4 × 1 is 4.
4
3.5
2
0
−2
−3.5
−4
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 4.
θ = 30°: turn 30°, go out 3.5.
And so on.
At 120°, r is negative: −2.
We turn 120°, then go 2 the other way, backwards through the center.
Careful!
A negative r is not nothing.
We go backwards, through the center, to the other side.
Careful: negative r: go the other way
4 Join the points
Join the points in order, with a smooth curve.
5 Name it
r = 4 cos θ is a circle through the center, with diameter 4.
So it is a circle.
r = 4 cos θ: a circle
6 Check
Let's check one angle we did not use: θ = 45°.
cos 45° is 0.71.
So r is 2.8: turn 45°, go out 2.8.
The point is on our curve.
Right!
θ = 45°: r = 4 × cos 45° = 4 × 0.71 ≈ 2.8
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.
Answer r = 4 cos θ: a circle
▶ Watch it solved
The question
Question: graph r = 3 + 2 cos θ.
θ is the angle we turn, r is how far we go out.
r = 3 + 2 cos θ
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.
So θ from 0° to 180° is enough: the rest is the mirror.
cos(−θ) = cos θ: the same r above and below the x-axis
2 Make a table
A table: 7 angles, 30° apart, from 0° to 180°.
The angles go along the top.
0°
30°
60°
90°
120°
150°
180°
cos of each angle, from the unit circle, to 2 decimal places: 1, 0.87, 0.5, and so on.
1
0.87
0.5
0
−0.5
−0.87
−1
Then r = 3 plus 2 times that.
For 0°: 3 + 2 × 1 is 5.
5
4.7
4
3
2
1.3
1
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 5.
θ = 30°: turn 30°, go out 4.7.
And so on.
Careful!
The point is not x = θ, y = r.
We turn θ, then walk out r along that line.
Careful: turn θ first, then go out r
4 Join and mirror
Join the points in order, with a smooth curve.
The other half is the mirror image in the x-axis.
5 Name it
3 is bigger than 2: r is never 0, so the curve never reaches the center.
It has a dent: a dimple.
So it is a limaçon.
r = 3 + 2 cos θ: a limaçon
6 Check
Let's check one angle we did not use: θ = 270°.
cos 270° is 0.
So r is 3: turn 270°, go out 3.
The point is on our curve.
Right!
θ = 270°: r = 3 + 2 × cos 270° = 3 + 2 × 0 = 3
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.
Answer r = 3 + 2 cos θ: a limaçon
▶ Watch it solved
The question
Question: graph r = 1 + 2 cos θ.
θ is the angle we turn, r is how far we go out.
r = 1 + 2 cos θ
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.
So θ from 0° to 180° is enough: the rest is the mirror.
cos(−θ) = cos θ: the same r above and below the x-axis
2 Make a table
A table: 7 angles, 30° apart, from 0° to 180°.
The angles go along the top.
0°
30°
60°
90°
120°
150°
180°
cos of each angle, from the unit circle, to 2 decimal places: 1, 0.87, 0.5, and so on.
1
0.87
0.5
0
−0.5
−0.87
−1
Then r = 1 plus 2 times that.
For 0°: 1 + 2 × 1 is 3.
3
2.7
2
1
0
−0.7
−1
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 3.
θ = 30°: turn 30°, go out 2.7.
And so on.
At 150°, r is negative: −0.7.
We turn 150°, 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.
Careful: negative r: go the other way
4 Join and mirror
Join the points in order, with a smooth curve.
The other half is the mirror image in the x-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.
r = 1 + 2 cos θ: a limaçon with an inner loop
6 Check
Let's check one angle we did not use: θ = 270°.
cos 270° is 0.
So r is 1: turn 270°, go out 1.
The point is on our curve.
Right!
θ = 270°: r = 1 + 2 × cos 270° = 1 + 2 × 0 = 1
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.
Answer r = 1 + 2 cos θ: a limaçon with an inner loop
▶ Watch it solved
