Polar Equations - Free Printable

The image shows a polar plot of the function:

$$
r = e^{\cos\theta} - 2\cos(4\theta) + \sin\left(\frac{\theta}{12}\right)^5
$$

This is a complex polar curve, and your task seems to be analyzing or interpreting this graph. Since no explicit question was asked (e.g., "find symmetry", "determine number of petals", "identify key features"), I'll analyze and explain the structure and behavior of this polar curve based on its equation and visual appearance.

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🔍 Step-by-Step Analysis

#### 1. Understanding the Components

The function has three terms:
$$
r(\theta) = e^{\cos\theta} - 2\cos(4\theta) + \left[\sin\left(\frac{\theta}{12}\right)\right]^5
$$

Let’s break it down:

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##### ✔ Term 1: $ e^{\cos\theta} $

- This is always positive since $ e^x > 0 $ for all real $ x $.
- $ \cos\theta $ ranges from $-1$ to $1$, so $ e^{\cos\theta} $ ranges from $ e^{-1} \approx 0.367 $ to $ e^{1} \approx 2.718 $.
- It's periodic with period $ 2\pi $, and modulates the overall "amplitude" of the curve.
- Smooth and symmetric about the x-axis.

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##### ✔ Term 2: $ -2\cos(4\theta) $

- This term has frequency 4 → it contributes 4 oscillations over $ [0, 2\pi] $.
- The amplitude is 2, and it’s negative, so it pulls $ r $ downward when $ \cos(4\theta) $ is positive.
- This creates petal-like structures, and since $ \cos(4\theta) $ has 4 maxima/minima per cycle, this term tends to produce 8 petals in total (because cosine is even, and the sign alternates).
- But note: $ \cos(4\theta) $ has period $ \frac{\pi}{2} $, so over $ [0, 2\pi] $, it completes 4 full cycles → 8 lobes (since each maximum/minimum contributes a lobe).

Wait — actually, $ \cos(4\theta) $ produces 8 petals only if combined properly with symmetry.

But more precisely:
For $ r = a\cos(n\theta) $, the number of petals is:
- $ n $ if $ n $ is odd,
- $ 2n $ if $ n $ is even.

Here, $ n = 4 $, so $ \cos(4\theta) $ would normally create 8 petals. But because we have $ -2\cos(4\theta) $, the shape is flipped, but still gives 8 petals.

However, in our case, this term is added to other functions, so the final shape is not purely a rose curve.

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##### ✔ Term 3: $ \left[\sin\left(\frac{\theta}{12}\right)\right]^5 $

- This is a very slowly varying function due to $ \frac{\theta}{12} $.
- $ \sin(\theta/12) $ has period $ 24\pi $, which is much larger than $ 2\pi $, so over one full rotation ($ \theta \in [0, 2\pi] $), this term changes very slowly.
- Raising it to the 5th power makes it small near zero and emphasizes the sign (odd power preserves sign).
- So this term adds a slowly evolving perturbation to the radius, slightly modifying the shape over multiple rotations.

But since $ \theta \in [0, 2\pi] $, $ \frac{\theta}{12} \in [0, \frac{\pi}{6}] \approx [0, 0.5236] $, so $ \sin(\theta/12) \in [0, 0.51] $, and $ [\sin(\theta/12)]^5 \in [0, 0.035] $ — a small correction.

So this term contributes a tiny wobble or asymmetry to the curve, especially noticeable in the outer regions.

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🌸 Visual Interpretation

From the image:

- The curve appears to have four large petals and four smaller ones, arranged symmetrically.
- The overall shape resembles a four-petal flower, but with some asymmetry and distortion.
- The largest petals are along the diagonals (like at $ \theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 $).
- There are smaller loops near the origin, possibly due to interference between the terms.
- The curve is not perfectly symmetric, likely due to the slow-varying $ \sin(\theta/12)^5 $ term introducing a slight phase shift or asymmetry.

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🧩 Key Observations

| Feature | Explanation |
|--------|-------------|
| Number of petals | Appears to have 4 major petals and 4 minor ones → total 8 lobes? But visually, it looks like 4 main petals with distortion. |
| Symmetry | Not fully symmetric — likely due to the $ \sin(\theta/12)^5 $ term breaking rotational symmetry slightly. |
| Origin crossing | Yes — $ r $ can be zero or negative depending on $ \theta $. When $ r < 0 $, the point plots in the opposite direction. |
| Dominant term | $ e^{\cos\theta} $ provides baseline magnitude; $ -2\cos(4\theta) $ drives petal formation; $ \sin(\theta/12)^5 $ adds fine detail. |

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💡 Conclusion: What Is This Curve?

This is a composite polar curve combining:
- An exponential modulation ($ e^{\cos\theta} $),
- A high-frequency oscillation ($ \cos(4\theta) $) creating petal-like structures,
- A low-frequency perturbation ($ \sin(\theta/12)^5 $) adding subtle asymmetry.

The resulting graph is a four-petaled flower with distorted symmetry, where:
- The main structure comes from $ -2\cos(4\theta) $,
- The modulation from $ e^{\cos\theta} $ controls how far out the petals extend,
- The tiny wobble from $ \sin(\theta/12)^5 $ causes slight asymmetry.

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✔ Final Answer (if the task was to interpret the graph):

> The given polar curve $ r = e^{\cos\theta} - 2\cos(4\theta) + \left[\sin\left(\frac{\theta}{12}\right)\right]^5 $ produces a four-petaled flower-like shape with slight asymmetry due to a slowly varying perturbation. The dominant features arise from the $ -2\cos(4\theta) $ term, which generates petal structures, while $ e^{\cos\theta} $ modulates the radial size, and $ \left[\sin(\theta/12)\right]^5 $ introduces a subtle distortion that breaks perfect symmetry. The graph exhibits rotational symmetry approximately every $ \pi/2 $ radians, but with small deviations due to the fifth-power sine term.

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If you had a specific question (e.g., "how many petals?", "find symmetry", "compute area", etc.), please clarify, and I’ll give a targeted answer!