SME
Grade 12 Math · Problem type
Limaçons and cardioids
Use this for “Find r on r = 2 + 2 cos θ at θ = 60°” and the curve’s shape.
Diagram
r = 2 + 2 cos θ: a cardioid, θ from 0° to 360°.
(r, θ) = (3, 60°)
This is an example. Type your own values to start a new problem.
r
=
a
+
b
cos
θ
°
Formulas
r
=
a
+
b
×
cos(
θ
°)
3 = 2 + 2 × cos(60°)
Assumptions
•
r = a + b cos θ: |a| = |b| is a cardioid, through the pole with a heart shape.
•
|a| < |b| has an inner loop; |a| > |b| has a dimple, or is convex when |a| ≥ 2|b|.
•
A negative r is plotted on the opposite ray.
Step-by-step
WE KNOW
a = 2, b = 2, θ = 60°
FIND
directed distance r (r)
Step 1 · Find directed distance r (r)
Use r = a + b × cos(θ°)
Put
θ
into
the
curve’s
equation.
r
=
2
+
2
×
cos(60°)
r
=
2
+
2
×
0.5
r
=
2
+
1
r
=
3
Check
Put the numbers back in. Both sides should match.
3 = 2 + 2 × cos(60°) ✓
Topic check
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Polar coordinates and polar form of complex numbers
Main lesson
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Use this for “Multiply 2(cos 30° + i sin 30°) by 3(cos 60° + i sin 60°).”
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Use this for “Graph r = 0.5θ: find r at θ = 270°” or “At what angle is r = 6?”