Objective: Demonstrate your understanding of limaçons by graphing polar equations and analyzing how the values of a and b affect the shape and type of the graph.

Question

Part A – Select and Graph Two Limaçons (3 points)
Choose one sine and one cosine equation with the same a and b values (a does not have to equal b).
Use desmos and the polar coordinate option to display your graph.
Label the graphs properly and display the equations.
Part B – Analyze Your Graphs (4 points) Results specific to their graphs.
Identify the type of limaçons for each graph (cardioid, inner loop, dimpled, or convex). Explain how you identified the type by your choices of a and b in your equation(s).
Explain how the values of a and b influenced the shape, orientation, and position relative to the pole for each graph.
Part C – Reflection (2 points) - Include at least 2 other limacon types for 2 points.
Discuss how changes in the ratio of a to b impact a limaçon’s appearance.
Submission of Work
Attach your equations and graphs.
Include your analysis and reflections as a written explanation.
Professional Work (1 point)

Answer 1

Sure! Below, I will outline a complete response as though you are engaging in the exercise. For this text, I'll provide both the equations for the limaçons, analyses, and reflections.

Part A – Select and Graph Two Limaçons

Equations:

• Cosine Limaçon: \( r = 2 + 2\cos(\theta) \)
• Sine Limaçon: \( r = 2 + 2\sin(\theta) \)

These equations can be graphed using a polar coordinate graphing tool like Desmos.

Graphs: where you can input the given equations to display them.

Label each graph clearly as "Limaçon - Cosine: \( r = 2 + 2\cos(\theta) \)" and "Limaçon - Sine: \( r = 2 + 2\sin(\theta) \)".

Part B – Analyze Your Graphs

Identifying the Type of Limaçons:

• Cosine Limaçon (r = 2 + 2cos(θ)): This graph resembles a convex limaçon. Since \( a = 2 \) and \( b = 2 \), the limaçon does not have an inner loop, as \( a > b \).

• Sine Limaçon (r = 2 + 2sin(θ)): This graph also resembles a convex limaçon for the same reasons.

How Values of a and b Influence the Shape:

1. Convex Limaçon: In both cases, since \( a = b \), which is equal and greater than 0, this results in both graphs being convex, meaning there is no inner loop and they have a smooth outer curve.
2. Orientation: The cosine version is symmetric about the polar axis (the horizontal axis), while the sine version is symmetric about the line \(\theta = \frac{\pi}{2}\) (the vertical axis). Thus, they differ in orientation despite the curves being similar in shape.
3. Position Relative to the Pole: Both limaçons have their closest points (minimum radius) at the pole when \(\theta = 0\) for cosine and \(\theta = \frac{\pi}{2}\) for sine, resulting in both having the same distance from the pole.

Part C – Reflection

Other Limaçon Types:

1. Cardioid Example (r = 1 + 1cos(θ)): Here, \( a = 1 \) and \( b = 1 \). This graph forms a cardioid, identifiable by the fact \( a = b \).
2. Limaçon with an Inner Loop (r = 1 + 2cos(θ)): In this case, \( a = 1, b = 2 \). When \( a < b \), you get a limaçon with an inner loop, which can be observed in its graph as it dips below the pole.

Impact of the Ratio of a to b:

• When \( a < b \), the limaçon has an inner loop, as it dips below the origin and then comes back up. This produces a fold in the graph where the radius can reach negative values, causing a loop.
• When \( a = b \), the limaçon turns into a cardioid, showcasing a single cusp point.
• When \( a > b \), the limaçon remains convex, as seen in both the cosine and sine examples provided. Here, the graph will have a fuller shape without any depressions below the origin.

Submission of Work

I would graphically display the polar equations using Desmos, label the graphs accordingly, and include my analyses and reflections based on the provided descriptions.

Conclusion

Through this exercise, it becomes apparent how crucial the ratios of \( a \) and \( b \) are in determining the specific type of limaçon produced, as well as its key characteristics such as orientation and proximity to the pole.

---

Feel free to input your equations into a graphing calculator and analyze them in the manner described for your assignment!