The conversion between Cartesian and polar coordinates is given by:

$x = r \cos(\theta)$

$y = r \sin(\theta)$

$$r = a(1 + \cos(\theta))$$

**Properties of a Cardioid:**

• It has a single cusp.
• The curve is symmetric about the polar axis.
• The maximum radial distance is $2a$, and the minimum is $0$.

**Example:** Graphing the equation $r = 2(1 + \cos(\theta))$ produces a cardioid with a maximum radius of $4$ units and a cusp at the origin.

$$r = a \cos(k\theta) \quad \text{or} \quad r = a \sin(k\theta)$$

**Determining the Number of Petals:**

• If $k$ is odd, the number of petals is $k$.
• If $k$ is even, the number of petals is $2k$.

**Examples:**

• For $r = 3 \cos(4\theta)$, since $k=4$ is even, there are $8$ petals.
• For $r = 2 \sin(3\theta)$, since $k=3$ is odd, there are $3$ petals.

**Applications of Rose Curves:**

• Modeling floral patterns and natural phenomena.
• Designing artistic and architectural elements.

$$r = a + b\theta$$

**Characteristics of the Archimedean Spiral:**

• The spiral increases its distance from the origin at a constant rate.
• It has an infinite number of turns.
• Widely used in various applications such as spiral staircases and galaxy formations.

**Example:**

Graphing $r = 1 + 0.5\theta$ produces an Archimedean spiral that starts at $r = 1$ when $\theta = 0$ and increases linearly as $\theta$ increases.

1. Identify the Type of Curve: Determine whether the equation represents a cardioid, rose, or spiral based on its form.
2. Determine Key Parameters: Find values of $a$, $b$, and $k$ to understand the size, number of petals, or the rate of spiraling.
3. Plot Key Points: Calculate $r$ for various angles $\theta$ to plot points on the polar grid.
4. Sketch the Curve: Connect the plotted points smoothly, respecting the symmetry and properties of the curve.
5. Analyze Symmetry: Many polar curves exhibit symmetry about the polar axis, the line $\theta = \frac{\pi}{2}$, or the origin.

**Example:** Graphing $r = 1 + \cos(\theta)$ involves identifying it as a cardioid, calculating $r$ for several $\theta$ values between $0$ and $2\pi$, plotting these points, and connecting them to form the heart-shaped curve.

• Engineering: Designing gears, turbines, and optical instruments.
• Biology: Modeling natural patterns such as seashells and flower petals.
• Art and Design: Creating aesthetically pleasing geometric patterns and structures.
• Astronomy: Describing the orbits of celestial bodies and spiral galaxies.

• Periodicity: Roses have periodicity depending on the value of $k$ in their equations.
• Symmetry: Cardioids are symmetric about the polar axis, while roses may have rotational symmetry.
• Asymptotes: Spirals like the Archimedean spiral do not have asymptotes as they extend infinitely.
• Intersection Points: Determining points where curves intersect requires solving polar equations simultaneously.

**Example:** To find the points of intersection between two rose curves $r = 2\cos(3\theta)$ and $r = 2\sin(3\theta)$, set the equations equal and solve for $\theta$:

$$2\cos(3\theta) = 2\sin(3\theta)$$

$$\cos(3\theta) = \sin(3\theta)$$

$$3\theta = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}$$

Thus, the intersection points occur at angles $\theta = \frac{\pi}{12} + \frac{n\pi}{3}$.

Area Enclosed by a Polar Curve:

$$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$$

Arc Length of a Polar Curve:

$$L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$$

**Example:** To find the area of one petal of the rose $r = 3\cos(2\theta)$:

Identify the interval for one petal, say $\theta \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

Calculate the area:

$$A = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (3\cos(2\theta))^2 d\theta = \frac{9}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(2\theta) d\theta$$

Using the identity $\cos^2 x = \frac{1 + \cos(2x)}{2}$:

$$A = \frac{9}{4} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (1 + \cos(4\theta)) d\theta = \frac{9}{4} \left[\theta + \frac{\sin(4\theta)}{4}\right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{9}{4} \cdot \frac{\pi}{2} = \frac{9\pi}{8}$$

The area of one petal is $\frac{9\pi}{8}$ square units.

• Polar coordinates provide a unique framework for graphing complex curves like cardioids, roses, and spirals.
• Cardioids feature a single cusp and are defined by $r = a(1 + \cos(\theta))$.
• Rose curves produce petal patterns influenced by the parameter $k$ in their equations.
• Spirals, such as the Archimedean spiral, extend infinitely with a linear increase in radius.
• Mastering graphing techniques and mathematical properties enhances understanding and application of these polar curves.