Limaçons and Cardioids
Curves like r = a + b cosθ.
Key idea
Curves like r = a + b cosθ.
In depth
r = a + b cos θ gives a limaçon. If a = b it is a heart-shaped cardioid; if a < b it has an inner loop; if a > b it is dimpled or convex. The ratio a/b controls the shape.
Worked example
r = 1 + cos θ is a cardioid, with a cusp at the origin.
Real-world uses
- Modelling microphone pick-up patterns and heart-shaped sensitivity regions in audio engineering
Common mistake
Plotting only a half period, which gives half the curve.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Limaçons and Cardioids?
Curves like r = a + b cosθ. It is a topic at the expert level in Trigonometry.
Where is Limaçons and Cardioids used in real life?
Real-world examples: Modelling microphone pick-up patterns and heart-shaped sensitivity regions in audio engineering.
What should I learn before Limaçons and Cardioids?
Make sure you are comfortable with Polar Graphs: Circles and Roses, the previous topic in Trigonometry.
What is a common mistake with Limaçons and Cardioids?
Plotting only a half period, which gives half the curve.
How can I practise Limaçons and Cardioids?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Limaçons and Cardioids?
Move on to Complex Numbers in Polar Form next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).