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The image shows a mathematical problem in Indonesian related to symmetry analysis of a polar equation, which states:

**Problem Statement:**
"Analisis kesimetrisan persamaan \( r^2 = 4\cos 2\theta \) dan sketsa grafiknya."

**Solution Approach:**
"Penyelesaian: Lakukan uji simetri untuk menganalisa kesimetrisan persamaan \( r^2 = 4\cos 2\theta \)."

This involves determining the symmetry of the given polar equation. Let's address the problem systematically:

### **Step 1: Symmetry Tests for Polar Equations**

To analyze symmetry in polar coordinates, there are three key types of symmetry to check:
1. **Symmetry about the Polar Axis (\( \theta = 0 \) or \( x \)-axis):** Replace \( \theta \) with \( -\theta \).
2. **Symmetry about the Line \( \theta = \frac{\pi}{2} \) (vertical or \( y \)-axis):** Replace \( \theta \) with \( \pi - \theta \).
3. **Symmetry about the Origin:** Replace \( r \) with \( -r \) or \( \theta \) with \( \theta + \pi \).

### **Applying the Tests to \( r^2 = 4\cos 2\theta \)**

#### **1. Test for Symmetry about the Polar Axis:**
   Replace \( \theta \) with \( -\theta \):
   \[
   r^2 = 4\cos(-2\theta).
   \]
   Since \( \cos(-2\theta) = \cos(2\theta) \), the equation remains unchanged. Thus, the equation is **symmetric about the polar axis**.

#### **2. Test for Symmetry about the Line \( \theta = \frac{\pi}{2} \):**
   Replace \( \theta \) with \( \pi - \theta \):
   \[
   r^2 = 4\cos(2(\pi - \theta)).
   \]
   This simplifies to:
   \[
   r^2 = 4\cos(2\pi - 2\theta) = 4\cos(2\theta),
   \]
   since \( \cos(2\pi - 2\theta) = \cos(2\theta) \). The equation remains unchanged, indicating **symmetry about the line \( \theta = \frac{\pi}{2} \)**.

#### **3. Test for Symmetry about the Origin:**
   Replace \( r \) with \( -r \):
   \[
   (-r)^2 = 4\cos 2\theta \implies r^2 = 4\cos 2\theta.
   \]
   The equation remains unchanged, indicating **symmetry about the origin**.

### **Graphical Representation:**
The polar equation \( r^2 = 4\cos 2\theta \) is likely to represent a lemniscate (a figure-eight shape), due to the presence of \( \cos 2\theta \) term.

Would you like further explanation or a sketch of the graph? 

Here are some related questions:
1. How do you determine the type of curve given a polar equation?
2. What is the significance of the \( \cos 2\theta \) term in the equation?
3. How would the graph change if the equation involved \( \sin 2\theta \) instead?
4. How does replacing \( r^2 \) with \( r \) affect the symmetry and graph?
5. Why do polar coordinates simplify symmetry analysis for certain curves?

**Tip:** When analyzing symmetry in polar equations, substituting values to check each symmetry can help you understand the curve’s orientation and properties effectively.

Analisis kesimetrisan persamaan r^2 = 4cos 2θ dan sketsa grafiknya.

Explore the symmetry analysis of the polar equation r^2 = 4cos 2θ and how to sketch its graph. This problem introduces key concepts in polar coordinates, symmetry, and trigonometric identities. Suitable for advanced students.

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