Unit 9.7 – Defining Polar Coordinates and Differentiating in Polar Form BC ONLY

AP® Calculus BC | Polar Coordinate System

Why This Matters: Polar coordinates provide an alternative to the Cartesian (x, y) system! Instead of horizontal and vertical distances, we use a distance from the origin (r) and an angle (θ). This system is perfect for describing circles, spirals, and rose curves. Polar calculus appears on every BC exam!

📍 What Are Polar Coordinates?

POLAR COORDINATE SYSTEM

A point in polar coordinates is represented as:

\[ (r, \theta) \]

Components:

\(r\): Distance from the origin (pole)
\(\theta\): Angle measured counterclockwise from the positive x-axis (polar axis)
\(r\) can be positive, negative, or zero
\(\theta\) is typically measured in radians

📝 Key Point: If \(r < 0\), the point is plotted in the opposite direction of angle \(\theta\). For example, \((-2, \pi/4)\) is the same as \((2, 5\pi/4)\).

🔄 Converting Between Polar and Rectangular

Conversion Formulas

FROM POLAR TO RECTANGULAR:

\[ x = r\cos\theta \]

\[ y = r\sin\theta \]

FROM RECTANGULAR TO POLAR:

\[ r = \sqrt{x^2 + y^2} \]

\[ \tan\theta = \frac{y}{x} \quad \text{(be careful with quadrant!)} \]

Or equivalently: \(r^2 = x^2 + y^2\)

⚠️ QUADRANT WARNING: When finding \(\theta\) from \(\tan\theta = y/x\), make sure to consider which quadrant the point is in! Use \(\arctan\) carefully or use \(\arctan2(y, x)\) function.

🌟 Common Polar Curves

Famous Polar Curves
Equation	Curve Name	Description
\(r = a\)	Circle	Circle centered at origin, radius \(a\)
\(r = a\cos\theta\) or \(r = a\sin\theta\)	Circle	Circle through origin
\(r = a + b\cos\theta\)	Limaçon	Has loop if \(\|a\| < \|b\|\)
\(r = a\cos(n\theta)\) or \(r = a\sin(n\theta)\)	Rose Curve	\(n\) petals if \(n\) odd, \(2n\) if even
\(r = a\theta\)	Spiral of Archimedes	Spiral outward
\(r^2 = a^2\cos(2\theta)\)	Lemniscate	Figure-eight shape

📐 Finding dy/dx in Polar Coordinates

The Derivative Formula

THE KEY FORMULA:

For a curve \(r = f(\theta)\):

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta} \]

Derivation:

Starting from \(x = r\cos\theta\) and \(y = r\sin\theta\):

\[ \frac{dx}{d\theta} = \frac{dr}{d\theta}\cos\theta - r\sin\theta \]

\[ \frac{dy}{d\theta} = \frac{dr}{d\theta}\sin\theta + r\cos\theta \]

Using the chain rule:

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

✨ Simplified Notation

Using Prime Notation:

Let \(r' = \frac{dr}{d\theta}\). Then:

\[ \frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta} \]

📏 Horizontal and Vertical Tangent Lines

Special Tangent Lines

Horizontal Tangent Lines:

Occur when \(\frac{dy}{dx} = 0\) (numerator = 0, denominator ≠ 0):

\[ \frac{dr}{d\theta}\sin\theta + r\cos\theta = 0 \]

Vertical Tangent Lines:

Occur when \(\frac{dy}{dx}\) is undefined (denominator = 0, numerator ≠ 0):

\[ \frac{dr}{d\theta}\cos\theta - r\sin\theta = 0 \]

📝 Special Case: If both numerator and denominator are 0, the tangent line may be indeterminate (use L'Hôpital's Rule or further analysis).

📖 Comprehensive Worked Examples

Example 1: Converting Coordinates

Problem: Convert the polar point \((4, \pi/3)\) to rectangular coordinates.

Solution:

Use conversion formulas:

\[ x = r\cos\theta = 4\cos\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} = 2 \]

\[ y = r\sin\theta = 4\sin\left(\frac{\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \]

ANSWER: \((2, 2\sqrt{3})\)

Example 2: Finding dy/dx

Problem: Find \(\frac{dy}{dx}\) for the curve \(r = 1 + \cos\theta\) at \(\theta = \pi/2\).

Step 1: Find \(\frac{dr}{d\theta}\)

\[ r = 1 + \cos\theta \quad \Rightarrow \quad \frac{dr}{d\theta} = -\sin\theta \]

Step 2: Apply the formula

\[ \frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta} \]

\[ = \frac{-\sin\theta \cdot \sin\theta + (1+\cos\theta)\cos\theta}{-\sin\theta \cdot \cos\theta - (1+\cos\theta)\sin\theta} \]

Step 3: Evaluate at \(\theta = \pi/2\)

At \(\theta = \pi/2\): \(\sin(\pi/2) = 1\), \(\cos(\pi/2) = 0\), \(r = 1\)

\[ \frac{dy}{dx} = \frac{-1 \cdot 1 + 1 \cdot 0}{-1 \cdot 0 - 1 \cdot 1} = \frac{-1}{-1} = 1 \]

Example 3: Horizontal Tangents

Problem: Find where \(r = 2\cos\theta\) has horizontal tangent lines.

Find \(r'\):

\[ r' = -2\sin\theta \]

Set numerator = 0:

\[ r'\sin\theta + r\cos\theta = 0 \]

\[ -2\sin\theta \cdot \sin\theta + 2\cos\theta \cdot \cos\theta = 0 \]

\[ -2\sin^2\theta + 2\cos^2\theta = 0 \]

\[ \cos^2\theta = \sin^2\theta \quad \Rightarrow \quad \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]

Check denominator ≠ 0 at these values

(Verify each value satisfies the condition)

Example 4: Converting Equation

Problem: Convert \(r = 4\sin\theta\) to rectangular form.

Multiply both sides by \(r\):

\[ r^2 = 4r\sin\theta \]

Using \(r^2 = x^2 + y^2\) and \(y = r\sin\theta\):

\[ x^2 + y^2 = 4y \]

\[ x^2 + y^2 - 4y = 0 \]

Complete the square:

\[ x^2 + (y-2)^2 = 4 \]

This is a circle with center \((0, 2)\) and radius 2!

💡 Essential Tips & Strategies

✅ Success Strategies:

Memorize conversion formulas: \(x = r\cos\theta\), \(y = r\sin\theta\)
For dy/dx: Find \(r'\) first, then use the formula
Horizontal tangents: Numerator = 0
Vertical tangents: Denominator = 0
Sketch curves: Helps visualize the problem
Check quadrants: When converting to polar
Simplify trig: Before evaluating derivatives
Calculator in radian mode: Always!

🔥 Common Polar Curves to Know:

Rose curves: \(r = a\cos(n\theta)\) → \(n\) or \(2n\) petals
Cardioid: \(r = a(1 \pm \cos\theta)\) or \(r = a(1 \pm \sin\theta)\)
Circles: \(r = a\), \(r = a\cos\theta\), \(r = a\sin\theta\)
Lemniscate: \(r^2 = a^2\cos(2\theta)\) or \(r^2 = a^2\sin(2\theta)\)

❌ Common Mistakes to Avoid

Mistake 1: Forgetting to multiply by \(r\) when converting equations
Mistake 2: Wrong quadrant when finding \(\theta\) from \(\tan\theta = y/x\)
Mistake 3: Not finding \(r'\) before using the derivative formula
Mistake 4: Confusing numerator/denominator conditions for horizontal/vertical tangents
Mistake 5: Calculator in degree mode instead of radians
Mistake 6: Sign errors in the derivative formula
Mistake 7: Not checking if denominator ≠ 0 for horizontal tangents
Mistake 8: Forgetting product rule when finding \(dx/d\theta\) and \(dy/d\theta\)
Mistake 9: Mixing up \(r\) and \(\theta\) in formulas
Mistake 10: Not simplifying trig expressions before evaluating

📝 Practice Problems

Solve these:

Convert \((2, \pi/6)\) from polar to rectangular.
Convert \((1, \sqrt{3})\) from rectangular to polar.
Find \(\frac{dy}{dx}\) for \(r = \sin\theta\) at \(\theta = \pi/4\).
Find horizontal tangents for \(r = 1 + \sin\theta\).

Answers:

\((\sqrt{3}, 1)\)
\((2, \pi/3)\) or equivalent
\(\frac{dy}{dx} = 1\)
\(\theta = \pi/6, 5\pi/6\)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Show conversion formulas: Write \(x = r\cos\theta\) explicitly
For derivatives: Show \(r'\) calculation
Show the full formula: Don't skip to the answer
For tangent lines: Set up the equation correctly
Check conditions: Denominator ≠ 0 for horizontal tangents
Simplify before evaluating: Makes calculation easier
Include proper notation: \(\theta\) vs \(x\)
State answers clearly: With proper form

💯 Exam Strategy:

Identify what's given: polar or rectangular?
For conversions: write formulas first
For derivatives: find \(r'\) immediately
Memorize the dy/dx formula
For tangent lines: know numerator/denominator conditions
Simplify trig before substituting values
Check your calculator mode!
Verify answers make sense

⚡ Quick Reference Guide

POLAR COORDINATES ESSENTIALS

Polar to Rectangular:

\(x = r\cos\theta\)
\(y = r\sin\theta\)

Rectangular to Polar:

\(r = \sqrt{x^2 + y^2}\)
\(\tan\theta = \frac{y}{x}\) (check quadrant!)

Derivative in Polar:

\[ \frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta} \]

Tangent Lines:

Horizontal: \(r'\sin\theta + r\cos\theta = 0\)
Vertical: \(r'\cos\theta - r\sin\theta = 0\)

Remember:

Find \(r' = dr/d\theta\) first!
Always use radians
Check quadrant for \(\theta\)

Master Polar Coordinates! Polar system uses (r, θ) where \(r\) = distance from origin, \(\theta\) = angle from positive x-axis. Conversions: polar to rectangular: \(x = r\cos\theta\), \(y = r\sin\theta\); rectangular to polar: \(r = \sqrt{x^2+y^2}\), \(\tan\theta = y/x\) (watch quadrant!). Derivative for curve \(r = f(\theta)\): \(\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}\) where \(r' = dr/d\theta\). Tangent lines: horizontal when numerator = 0, vertical when denominator = 0. Common polar curves: circles (\(r = a\)), rose curves (\(r = a\cos(n\theta)\)), cardioids (\(r = a(1±\cos\theta)\)), lemniscates (\(r^2 = a^2\cos(2\theta)\)). Critical: always use radians, find \(r'\) first for derivatives, check quadrants for angle. This is guaranteed BC content—appears every year! Practice conversions and derivatives until automatic! 🎯✨