🎯⭐ INTERACTIVE LESSON

Polar Coordinates

Learn step-by-step with interactive practice!

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Polar Coordinates - Complete Interactive Lesson

Part 1: Polar Coordinate System

📍 Introduction to Polar Coordinates

Part 1 of 7

Instead of locating points by horizontal/vertical distances $(x, y)$ , polar coordinates use a distance and angle: $(r, \theta)$ .

Polar vs. Rectangular

Coordinate System	Point Defined By	Notation
Rectangular (Cartesian)	Horizontal & vertical distances	$(x, y)$
Polar	Distance from origin & angle from positive $x$ -axis	$(r, \theta)$

Key Components

$r$ = distance from the pole (origin)
$\theta$ = angle measured counterclockwise from the polar axis (positive $x$ -axis)

$\boxed{\text{Point } (r, \theta): \text{ go distance } r \text{ in direction } \theta}$

Important Notes

$r$ can be negative: $(- r, \theta)$ means go distance $r$ in the opposite direction of $\theta$
Angles can exceed $360°$ or be negative (clockwise)

🔄 Converting Between Systems

Polar → Rectangular

$\boxed{x = r\cos\theta, \qquad y = r\sin\theta}$

Rectangular → Polar

🔁 Multiple Representations

The Same Point Has Many Names

The point $(r, \theta)$ is also represented by:

$(r, \theta + 2\pi n)$ for any integer $n$
for any integer

Polar Basics Quiz 🎯

Convert Coordinates 🧮

1) Convert $(5, \frac{\pi}{6})$ to rectangular. What is $x$ ? Round to 1 decimal. (e.g., $4\cos\frac{\pi}{3} = 4(0.5) = 2.0$ )

Coordinate Matching 🔽

Exit Quiz ✅

Part 2: Converting Coordinates

🌹 Polar Curves — Basic Shapes

Part 2 of 7

Polar equations define curves using $r$ as a function of $\theta$ . The shapes are often strikingly beautiful.

Lines & Circles in Polar

Equation	Shape
$\theta = c$	Line through origin at angle $c$

Part 3: Polar Graphs

🔄 Converting Polar ↔ Rectangular Equations

Part 3 of 7

Converting equations between polar and rectangular form is essential for graphing and analysis.

Key Substitution Relationships

Polar → Rectangular	Rectangular → Polar
$r\cos\theta = x$	$x = r\cos\theta$

Part 4: Rose Curves & Limacons

📊 Graphing Polar Equations by Hand

Part 4 of 7

Graphing polar equations by hand requires building a table of $(r, \theta)$ values and plotting points.

Step-by-Step Process

Make a table of $\theta$ values (usually multiples of $\frac{\pi}{6}$ or )

Part 5: Polar Equations

📐 Area in Polar Coordinates

Part 5 of 7

The Polar Area Formula

To find the area enclosed by a polar curve $r = f(\theta)$ from $\theta = \alpha$ to $\theta = \beta$ :

Part 6: Problem-Solving Workshop

🪐 Conic Sections in Polar Form

Part 6 of 7

The Focus-Directrix Form

Any conic section (ellipse, parabola, hyperbola) with one focus at the origin can be written:

$r = \frac{ed}{1 \pm e\cos\theta} \quad \text{or} \quad r = \frac{ed}{1 \pm e\sin\theta}$

Part 7: Review & Applications

🧩 Polar Coordinates — Full Synthesis

Part 7 of 7

Everything Together

This final part combines all polar coordinate skills:

Skill	Key Formula / Concept
Conversions	$x = r\cos\theta, \; y = r\sin\theta, \; r^2 = x^2+y^2$

The same point has infinitely many polar representations

\boxed{r = \sqrt{x^2 + y^2}, \qquad \tan\theta = \frac{y}{x}}

When finding $\theta$ , always check the quadrant — $\arctan$ alone may give the wrong angle!

Example 1: Polar → Rectangular

Convert $(4, \frac{\pi}{3})$ to rectangular:

$x = 4\cos\frac{\pi}{3} = 4 \cdot \frac{1}{2} = 2$

$y = 4\sin\frac{\pi}{3} = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$

Answer: $(2, 2\sqrt{3})$

Example 2: Rectangular → Polar

Convert $(-3, 3)$ to polar:

$r = \sqrt{9 + 9} = 3\sqrt{2}$

$\tan\theta = \frac{3}{-3} = -1$ . Since the point is in QII: $\theta = \frac{3\pi}{4}$

Answer: $(3\sqrt{2}, \frac{3\pi}{4})$

(- r, θ + π + 2 πn)

Example: All Representations of $(3, \frac{\pi}{4})$

$(3, \frac{\pi}{4})$ — standard
$(3, \frac{9\pi}{4})$ — add $2\pi$
$(-3, \frac{5\pi}{4})$ — negate $r$ and add $\pi$
$(-3, -\frac{3\pi}{4})$ — negate $r$ and subtract $\pi$

Plotting Negative $r$

To plot $(-2, \frac{\pi}{6})$ :

Face direction $\frac{\pi}{6}$ (30°)
Walk backwards 2 units
You end up at $(2, \frac{7\pi}{6})$ — same point!

4 cos \frac{π}{3} = 4 (0.5) = 2.0

2) Convert $(5, \frac{\pi}{6})$ to rectangular. What is $y$ ? (e.g., $4\sin\frac{\pi}{3} = 4(0.866) = 3.5$ )

3) Convert rectangular $(0, -4)$ to polar. What is $r$ ? (e.g., $\sqrt{3^2 + 4^2} = 5$ )

Example: $r = 4\cos\theta$

This is a circle with diameter $4$ , centered at $(2, 0)$ in rectangular coordinates.

To verify: $r = 4\cos\theta \implies r^2 = 4r\cos\theta \implies x^2 + y^2 = 4x$

$\implies (x-2)^2 + y^2 = 4$

Circle with center $(2, 0)$ and radius $2$ . ✓

🌹 Rose Curves

Standard Forms

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

$n$	# of Petals
Odd $n$	$n$ petals
Even $n$	$2n$ petals

Petal length = $|a|$

Examples

Equation	Petals	Petal Length
$r = 3\cos(2\theta)$	$4$ petals	$3$
$r = 5 \sin ($

Why the Odd/Even Rule?

For odd $n$ : each petal is traced once as $\theta$ goes from $0$ to $\pi$ .

For even $n$ : petals in each "half" are traced, and the curve also traces petals when $r$ is negative, doubling the count.

🐌 Limaçons

Standard Forms

$r = a \pm b\cos\theta \quad \text{or} \quad r = a \pm b\sin\theta$

The shape depends on the ratio $\frac{a}{b}$ :

Ratio	Shape
$\frac{a}{b} < 1$	Inner loop
$\frac{a}{b} = 1$

Example: $r = 2 + 3\cos\theta$ (Inner Loop)

$\frac{a}{b} = \frac{2}{3} < 1$ →

Maximum $r$ : when $\cos\theta = 1$ , $r = 5$
Minimum $r$ : when $\cos\theta = -1$ , (inner loop!)

Example: $r = 3 + 3\sin\theta$ (Cardioid)

$\frac{a}{b} = 1$ → cardioid

Passes through the origin when $\sin\theta = -1$ , i.e., $\theta = \frac{3\pi}{2}$ .

Polar Curves Quiz 🎯

Polar Curve Analysis 🧮

1) $r = 6\cos(2\theta)$ : how many petals? (e.g., $r = a\cos(3\theta)$ has 3 petals since 3 is odd)

2) $r = 1 + 3\sin\theta$ : compute $\frac{a}{b}$ as a decimal. (e.g., for $r = 2 + 4\cos\theta$ : $\frac{a}{b} = \frac{2}{4} = 0.5$ )

3) $r = 4\sin\theta$ is a circle with what diameter? (e.g., $r = 6\cos\theta$ is a circle with diameter 6)

Curve Identification 🔽

Strategy: Polar → Rectangular

Look for $r\cos\theta$ (replace with $x$ ) or $r\sin\theta$ (replace with $y$ )
Look for $r^2$ (replace with $x^2 + y^2$ )
Multiply both sides by $r$ if needed to create these forms

📝 Converting Polar → Rectangular

Example 1: $r = 3$

$r^2 = 9 \implies x^2 + y^2 = 9$

Circle of radius 3.

Example 2: $r = 4\sec\theta$

$r = \frac{4}{\cos\theta} \implies r\cos\theta = 4 \implies x = 4$

Vertical line!

Example 3: $r = 2\sin\theta + 4\cos\theta$

Multiply by $r$ : $r^2 = 2r\sin\theta + 4r\cos\theta$

$x^2 + y^2 = 2y + 4x$

$(x-2)^2 + (y-1)^2 = 5$

Circle with center $(2, 1)$ and radius $\sqrt{5}$ .

Example 4: $r = \frac{6}{2\cos\theta + 3\sin\theta}$

$r(2\cos\theta + 3\sin\theta) = 6 \implies 2x + 3y = 6$

A straight line! In standard form: $2x + 3y = 6$ .

📝 Converting Rectangular → Polar

Example 5: $x^2 + y^2 = 16$

$r^2 = 16 \implies r = 4$

Example 6: $y = x$

$r\sin\theta = r\cos\theta \implies \tan\theta = 1 \implies \theta = \frac{\pi}{4}$

Example 7: $x^2 + y^2 - 6x = 0$

$r^2 - 6r\cos\theta = 0 \implies r(r - 6\cos\theta) = 0$

Since $r = 0$ is just the origin (already on the curve): $r = 6\cos\theta$

Example 8: $y = 3$

$r\sin\theta = 3 \implies r = 3\csc\theta$

Quick Reference

Rectangular	Polar
$x^2 + y^2 = a^2$

Conversion Quiz 🎯

Convert Equations 🧮

1) Convert $r = 8\cos\theta$ to rectangular. What is the radius of the resulting circle? (e.g., $r = 6\cos\theta \to (x-3)^2 + y^2 = 9$ , radius = 3)

2) Convert $x^2 + y^2 = 49$ to polar. What is $r$ ? (e.g., $x^2 + y^2 = 16$ becomes )

3) Convert $r = 3\sec\theta$ to rectangular. What is the constant $x$ value? (e.g., $r = 5\sec\theta \to x = 5$ )

Match the Forms 🔽

Compute

r

for each

\theta

Plot each

(r, \theta)

point on polar grid

Connect points with a smooth curve

Check symmetry to reduce work

Symmetry	Test	Replace
Polar axis ( $x$ -axis)	Replace $\theta$ with $-\theta$	If same equation: symmetric
Line $\theta = \frac{\pi}{2}$ ( $y$ -axis)	Replace $\theta$ with $\pi - \theta$	If same equation: symmetric
Pole (origin)	Replace $r$ with $-r$	If same equation: symmetric

📝 Graphing $r = 2 + 2\cos\theta$ (Cardioid)

Step 1: Table of Values

$\theta$	$\cos\theta$	$r = 2 + 2\cos\theta$
$0$	$1$	$4$
$\frac{\pi}{3}$

Step 2: Symmetry Check

Replace $\theta$ with $-\theta$ : $r = 2 + 2\cos(-\theta) = 2 + 2\cos\theta$ ✓

Symmetric about the polar axis! Only need to plot $[0, \pi]$ and reflect.

Key Feature

Passes through origin at $\theta = \pi$ (where $r = 0$ ).

🌹 Graphing a Rose: $r = 3\cos(2\theta)$

Finding the Petals

Set $r = 0$ : $\cos(2\theta) = 0 \implies 2\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$

$\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$

These are the "zeros" between petals.

Petal Locations

Maximum $r = 3$ when $\cos(2\theta) = 1$ :

$2\theta = 0, 2\pi, 4\pi \implies \theta = 0, \pi$ (petals along $x$ -axis)

Also $r = -3$ when $\cos(2\theta) = -1$ :

$2\theta = \pi, 3\pi \implies \theta = \frac{\pi}{2}, \frac{3\pi}{2}$ (petals along -axis, traced with negative )

Result

4 petals along the axes, each of length 3. The curve has both $x$ -axis and $y$ -axis symmetry, plus origin symmetry.

Graphing Quiz 🎯

Compute $r$ Values 🧮

For $r = 3 - 3\sin\theta$ :

1) At $\theta = 0$ : $r$ = ? (e.g., for $r = 2 + 2\cos\theta$ at $\theta = 0$ : $r = 2+2(1) = 4$ )

2) At $\theta = \frac{\pi}{2}$ : $r$ = ? (e.g., at $\theta = \frac{\pi}{2}$ : )

3) At $\theta = \frac{3\pi}{2}$ : $r$ = ? (e.g., at $\theta = \pi$ : )

Identify Features 🔽

$A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 \, d\theta = \frac{1}{2}\int_{\alpha}^{\beta} [f(\theta)]^2 \, d\theta$

Why $\frac{1}{2}r^2$ ?

Think of thin "pie slices" of angle $d\theta$ . Each slice is approximately a sector of a circle with area $\frac{1}{2}r^2 d\theta$ .

Identify the limits $\alpha$ and $\beta$ carefully
For a full curve, determine the period (e.g., a rose may complete in $[0, \pi]$ or $[0, 2\pi]$ )
Use symmetry to simplify: compute part and multiply

📝 Example: Area Inside a Cardioid $r = 1 + \cos\theta$

The full cardioid is traced from $\theta = 0$ to $\theta = 2\pi$ .

$A = \frac{1}{2}\int_0^{2\pi}(1+\cos\theta)^2\,d\theta$

Expand: $(1+\cos\theta)^2 = 1 + 2\cos\theta + \cos^2\theta$

Use $\cos^2\theta = \frac{1+\cos 2\theta}{2}$ :

$= 1 + 2\cos\theta + \frac{1}{2} + \frac{\cos 2\theta}{2} = \frac{3}{2} + 2\cos\theta + \frac{\cos 2\theta}{2}$

$A = \frac{1}{2}\int_0^{2\pi}\left(\frac{3}{2} + 2\cos\theta + \frac{\cos 2\theta}{2}\right)d\theta$

$= \frac{1}{2}\left[\frac{3}{2}\theta + 2\sin\theta + \frac{\sin 2\theta}{4}\right]_0^{2\pi} = \frac{1}{2}\left(\frac{3}{2}\cdot 2\pi\right) = \frac{3\pi}{2}$

Shortcut: By symmetry about the polar axis, we could compute $2 \cdot \frac{1}{2}\int_0^{\pi}(1+\cos\theta)^2\,d\theta$ , getting the same answer.

🔄 Area Between Polar Curves

For the area inside $r_1 = f(\theta)$ and outside $r_2 = g(\theta)$ (where $f(\theta) \geq g(\theta)$ ):

$A = \frac{1}{2}\int_{\alpha}^{\beta}\left([f(\theta)]^2 - [g(\theta)]^2\right)d\theta$

Example: Area inside $r = 2$ but outside $r = 2(1 - \cos\theta)$ .

Find intersections: $2 = 2(1-\cos\theta) \implies \cos\theta = 0 \implies \theta = \pm\frac{\pi}{2}$

By symmetry (both curves are symmetric about polar axis):

$A = 2 \cdot \frac{1}{2}\int_0^{\pi/2}\left(4 - 4(1-\cos\theta)^2\right)d\theta$

Carefully evaluate: $4 - 4(1-2\cos\theta+\cos^2\theta) = 8\cos\theta - 4\cos^2\theta$

This yields $A = 8 - \pi$ after integration.

⚠️ Common Mistake: Always check which curve is "outer" vs "inner" on the integration interval!

Set Up Area Integrals 🧮

1) Area inside $r = 4\sin\theta$ . This is a circle of diameter 4. Its area = ? (Enter as a multiple of $\pi$ , like "4pi")

2) One petal of $r = 3\cos(3\theta)$ : first petal from $\theta = 0$ to $\theta =$ ? (Enter as a fraction of pi, like "pi/3")

3) Area of one petal of $r = 2\sin(2\theta)$ : $A = \frac{1}{2}\int_0^{\pi/2} 4\sin^2(2\theta)\,d\theta =$ ? (Enter as a multiple of , like "pi/2")

Area Concepts 🔽

$e$ = eccentricity (determines shape)
$d$ = distance from focus to directrix

Classification by Eccentricity

Eccentricity	Conic Type
$e = 0$	Circle
$0 < e < 1$	Ellipse
$e = 1$	Parabola
$e > 1$	Hyperbola

$1 + e\cos\theta$ : directrix to the right of focus
$1 - e\cos\theta$ : directrix to the left of focus
$1 + e\sin\theta$ : directrix above focus
$1 - e\sin\theta$ : directrix below focus

📝 Example: Identify and Analyze $r = \frac{6}{2 + \cos\theta}$

Step 1: Standard Form

Divide numerator and denominator by 2: $r = \frac{3}{1 + \frac{1}{2}\cos\theta}$

So $e = \frac{1}{2}$ and $ed = 3 \implies d = 6$ .

Step 2: Classify

$e = \frac{1}{2} < 1$ → Ellipse

Step 3: Key Points

At $\theta = 0$ : $r = \frac{6}{2+1} = 2$ (closest to directrix)

Step 4: Semi-major axis

$a = \frac{r_{\min}+r_{\max}}{2} = \frac{2+6}{2} = 4$

Center is at distance $ae = 4 \cdot \frac{1}{2} = 2$ from the focus (origin).

🎯 Special Case: Parabola ( $e = 1$ )

$r = \frac{d}{1 + \cos\theta}$

At $\theta = 0$ : $r = \frac{d}{2}$ (vertex)
At $\theta = \frac{\pi}{2}$ : (end of latus rectum)

Latus rectum: The chord through the focus perpendicular to the axis has length $2d$ .

Converting to Rectangular

$r = \frac{d}{1+\cos\theta} \implies r(1+\cos\theta) = d \implies r + x = d$

$\sqrt{x^2+y^2} = d - x \implies x^2+y^2 = d^2 - 2dx + x^2 \implies y^2 = -2dx + d^2$

This is a parabola opening leftward!

Conic Classification 🎯

Analyze Conics 🧮

For $r = \frac{12}{3 + \cos\theta}$ :

1) Divide to standard form. The eccentricity $e$ = ? (Enter as a fraction like "1/3")

2) What is $r$ at $\theta = 0$ ? (Enter a whole number)

3) What is $r$ at $\theta = \pi$ ? (Enter a whole number)

Conic Properties 🔽

🎓 Problem-Solving Strategies

Identifying a Polar Curve

Flowchart:

$r = a$ → Circle centered at origin, radius $a$
$r = a\cos\theta$ or $r = a\sin\theta$ → Circle, diameter $|a|$
$r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$ → Limaçon
- : cardioid (passes through origin)
$r = a\cos(n\theta)$ or $r = a\sin(n\theta)$ → Rose
- $n$ odd: petals
$r^2 = a^2\cos(2\theta)$ or $r^2 = a^2\sin(2\theta)$ → Lemniscate (figure-8)
$r = \frac{ed}{1 \pm e\cos\theta}$ → Conic

Common Errors to Avoid

Forgetting squaring in area formula: it's $r^2$ , not $r$
Wrong limits: always find where $r = 0$ or where curves intersect
Negative $r$ : polar curves can overlap themselves when

Mixed Problems 🎯

Mixed Calculations 🧮

1) Convert $(x, y) = (-3, 3)$ to polar. What is $r$ ? (Enter exact value like "3sqrt2")

2) For $r = \frac{10}{5-3\cos\theta}$ , what is the eccentricity? (Enter as a fraction)

3) How many petals does $r = 4\sin(5\theta)$ have?

Exit Quiz — Final ✅

([f(θ)]2−[g(θ)]2)

\theta = \pi

r = \frac{6}{2-1} = 6

(farthest)

\theta = \frac{\pi}{2}

r = \frac{6}{2} = 3

\theta = \pi

: undefined (approaches infinity — the curve opens left)

a > b

: dimpled or convex limaçon (no inner loop)

a < b

: limaçon with inner loop

n

even:

2n

petals

Rectangular conversion:

\tan\theta = \frac{y}{x}

only in the correct quadrant

Polar Coordinates

Polar Coordinates - Complete Interactive Lesson

Part 1: Polar Coordinate System

📍 Introduction to Polar Coordinates

Polar vs. Rectangular

Key Components

Important Notes

🔄 Converting Between Systems

Polar → Rectangular

Rectangular → Polar

🔁 Multiple Representations

The Same Point Has Many Names

Part 2: Converting Coordinates

🌹 Polar Curves — Basic Shapes

Lines & Circles in Polar

Part 3: Polar Graphs

🔄 Converting Polar ↔ Rectangular Equations

Key Substitution Relationships

Part 4: Rose Curves & Limacons

📊 Graphing Polar Equations by Hand

Step-by-Step Process

Part 5: Polar Equations

📐 Area in Polar Coordinates

The Polar Area Formula

Part 6: Problem-Solving Workshop

🪐 Conic Sections in Polar Form

The Focus-Directrix Form

Part 7: Review & Applications

🧩 Polar Coordinates — Full Synthesis

Everything Together

Example 1: Polar → Rectangular

Example 2: Rectangular → Polar

Example: All Representations of (3,π4)(3, \frac{\pi}{4})(3,4π​)

Plotting Negative rrr

Example: r=4cos⁡θr = 4\cos\thetar=4cosθ

🌹 Rose Curves

Standard Forms

Examples

Why the Odd/Even Rule?

🐌 Limaçons

Standard Forms

Example: r=2+3cos⁡θr = 2 + 3\cos\thetar=2+3cosθ (Inner Loop)

Example: r=3+3sin⁡θr = 3 + 3\sin\thetar=3+3sinθ (Cardioid)

Strategy: Polar → Rectangular

📝 Converting Polar → Rectangular

Example 1: r=3r = 3r=3

Example 2: r=4sec⁡θr = 4\sec\thetar=4secθ

Example 3: r=2sin⁡θ+4cos⁡θr = 2\sin\theta + 4\cos\thetar=2sinθ+4cosθ

Example 4: r=62cos⁡θ+3sin⁡θr = \frac{6}{2\cos\theta + 3\sin\theta}r=2cosθ+3sinθ6​

📝 Converting Rectangular → Polar

Example 5: x2+y2=16x^2 + y^2 = 16x2+y2=16

Example 6: y=xy = xy=x

Example 7: x2+y2−6x=0x^2 + y^2 - 6x = 0x2+y2−6x=0

Example 8: y=3y = 3y=3

Quick Reference

Symmetry Tests

📝 Graphing r=2+2cos⁡θr = 2 + 2\cos\thetar=2+2cosθ (Cardioid)

Step 1: Table of Values

Step 2: Symmetry Check

Key Feature

🌹 Graphing a Rose: r=3cos⁡(2θ)r = 3\cos(2\theta)r=3cos(2θ)

Finding the Petals

Petal Locations

Result

Why 12r2\frac{1}{2}r^221​r2?

Key Setup Steps

📝 Example: Area Inside a Cardioid r=1+cos⁡θr = 1 + \cos\thetar=1+cosθ

🔄 Area Between Polar Curves

Classification by Eccentricity

Orientation

📝 Example: Identify and Analyze r=62+cos⁡θr = \frac{6}{2 + \cos\theta}r=2+cosθ6​

Step 1: Standard Form

Step 2: Classify

Step 3: Key Points

Step 4: Semi-major axis

🎯 Special Case: Parabola (e=1e = 1e=1)

Converting to Rectangular

🎓 Problem-Solving Strategies

Identifying a Polar Curve

Common Errors to Avoid

Example: All Representations of $(3, \frac{\pi}{4})$

Plotting Negative $r$

Example: $r = 4\cos\theta$

Example: $r = 2 + 3\cos\theta$ (Inner Loop)

Example: $r = 3 + 3\sin\theta$ (Cardioid)

Example 1: $r = 3$

Example 2: $r = 4\sec\theta$

Example 3: $r = 2\sin\theta + 4\cos\theta$

Example 4: $r = \frac{6}{2\cos\theta + 3\sin\theta}$

Example 5: $x^2 + y^2 = 16$

Example 6: $y = x$

Example 7: $x^2 + y^2 - 6x = 0$

Example 8: $y = 3$

📝 Graphing $r = 2 + 2\cos\theta$ (Cardioid)

🌹 Graphing a Rose: $r = 3\cos(2\theta)$

Why $\frac{1}{2}r^2$ ?

📝 Example: Area Inside a Cardioid $r = 1 + \cos\theta$

📝 Example: Identify and Analyze $r = \frac{6}{2 + \cos\theta}$

🎯 Special Case: Parabola ( $e = 1$ )