Section 8.2: Polar Coordinates Study Notes

Overview of the Cartesian and Polar Coordinate Systems

• Cartesian Coordinate System: The coordinate system most commonly used, consisting of a rectangular plane divided into four quadrants by a horizontal axis ($x$-axis) and a vertical axis ($y$-axis).
• Polar Coordinate System: An alternative way of describing a point's location, which can be more useful in certain contexts, particularly when the point's distance from the origin and its angle relative to the horizontal axis are the defining characteristics.

Understanding Polar Coordinates $(r, \theta)$

• Definition: Polar coordinates consist of an ordered pair $(r, \theta)$.     * $r$: Represents the distance from the point to the origin.     * $\theta$: Represents the angle measured in standard position (starting from the positive horizontal axis).
• Polar Grid Structure: If the plane were to be "gridded" for polar coordinates, it would consist of:     * Concentric circles at incremental radii.     * Rays drawn at incremental angles from the origin.

Plotting Points in Polar Coordinates

• Example 1: Plot the polar point $(3, \frac{5\pi}{6})$.     * This point is located at a distance of $3$ from the origin.     * The angle is $\frac{5\pi}{6}$.
• Example 2: Plot the polar point $(-2, \frac{\pi}{4})$.     * Typically, positive $r$ values are used, but they can be negative.     * To plot: First rotate to the angle $\frac{\pi}{4}$. Moving in this direction (into the first quadrant) represents positive $r$ values.     * For a negative $r$ ($-2$), move in the opposite direction from the angle $\frac{\pi}{4}$, which leads into the third quadrant.

Multiple Representations of Polar Points

• Non-Uniqueness: Unlike Cartesian coordinates, which have a unique pair for every point, a single Cartesian point can be represented by an infinite number of different polar coordinates.
• Alternative representations for the point from Example 2:     * The point $(-2, \frac{\pi}{4})$ results in the same location as $(2, \frac{5\pi}{4})$.     * Adding or subtracting full rotations ($2\pi$) to the angle results in the same point. For example, $(2, \frac{13\pi}{4})$ is also the same point.

Converting Between Polar and Cartesian Coordinates

• Fundamental Relationships: Conversion is based on the following trigonometric and algebraic relationships:     * $\cos(\theta) = \frac{x}{r} \implies x = r \times \cos(\theta)$     * $\sin(\theta) = \frac{y}{r} \implies y = r \times \sin(\theta)$     * $\tan(\theta) = \frac{y}{x}$     * $x^2 + y^2 = r^2$
• Unit Circle Connection: Knowledge of the unit circle is essential. For example, if $r = 1$ and $\theta = \frac{\pi}{3}$, the polar coordinates are $(1, \frac{\pi}{3})$ and the corresponding Cartesian coordinates are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

Examples of Point Conversion

• Example 3 (Polar to Cartesian): Find the Cartesian coordinates for $(r, \theta) = (5, \frac{2\pi}{3})$.     * $x = 5 \times \cos(\frac{2\pi}{3}) = 5 \times (-\frac{1}{2}) = -\frac{5}{2}$     * $y = 5 \times \sin(\frac{2\pi}{3}) = 5 \times (\frac{\sqrt{3}}{2}) = \frac{5\sqrt{3}}{2}$     * The Cartesian coordinates are $(-\frac{5}{2}, \frac{5\sqrt{3}}{2})$.
• Example 4 (Cartesian to Polar): Find the polar coordinates for $(x, y) = (-3, -4)$.     * Distance calculation: $r^2 = (-3)^2 + (-4)^2 = 9 + 16 = 25$, so $r = 5$.     * Angle calculation (using cosine): $\cos(\theta) = \frac{x}{r} = \frac{-3}{5}$.     * Solving for $\theta$: $\theta = \cos^{-1}(-\frac{3}{5}) \approx 2.214$     * The cosine relationship produces two solutions on the circle. By symmetry, the second possibility is $\theta = 2\pi - 2.214 = 4.069$.     * Choosing the quadrant: Since the Cartesian point $(-3, -4)$ is in the 3rd quadrant, the second angle is correct. The polar coordinates are $(5, 4.069)$.

Introduction to Polar Equations

• Definition: A polar equation describes a relationship between $r$ (radius) and $\theta$ (angle) on a polar grid, similar to how $y = f(x)$ works on a Cartesian grid.
• Example 5 (Spiral): Graphing $r = \theta$.     * This equation means the radius equals the angle.     * Table of values:         * $\theta = 0, r = 0$         * $\theta = \frac{\pi}{4}, r = \frac{\pi}{4}$         * $\theta = \frac{\pi}{2}, r = \frac{\pi}{2}$         * $\theta = \pi, r = \pi$         * $\theta = 2\pi, r = 2\pi$     * The resulting graph is a spiral. It fails the vertical line test in Cartesian coordinates, meaning it is not a function of the form $y = f(x)$.

Common Polar Graphs: Circles and Spirals

• Example 6 (Circle centered at origin): $r = 3$.     * Because $\theta$ is missing, the radius is always 3 regardless of the angle. This produces a circle with radius 3.
• Graphing Technology: Most calculators treat $y$ as a function of $x$. To graph polar equations, the mode must be changed to "Polar," where the utility asks for $r(\theta)$ or $r =$.
• Example 7 (Circle through origin): $r = 4 \times \cos(\theta)$.     * Technology shows this is a circle passing through the origin.     * Finding the interval for one cycle: Solve for when the graph returns to its starting point. At $\theta = 0$, $r = 4 \times \cos(0) = 4$. To find when it returns to $(4, 0)$, we solve $r \times \cos(\theta) = 4$.     * $4 \times \cos(\theta) \times \cos(\theta) = 4 \implies \cos^2(\theta) = 1$.     * Solutions: $\theta = 0$ or $\theta = \pi$.     * The interval 0 \le \theta < \pi completes one iteration.

Analyzing Complex Polar Curves: Limaçons and Roses

• Example 8 (Limaçon): $r = 4 \times \sin(\theta) + 2$.     * This graph features an inner loop.     * The inner loop starts and ends at the origin ($r = 0$).     * Solving $0 = 4 \times \sin(\theta) + 2 \implies \sin(\theta) = -\frac{1}{2}$.     * Solutions on $[0, 2\pi)$: $\theta = \frac{7\pi}{6}$ and $\theta = \frac{11\pi}{6}$.     * The inner loop corresponds to the interval $\frac{7\pi}{6} \le \theta \le \frac{11\pi}{6}$, where $r$ takes on negative values.
• Example 9 (Rose): $r = \cos(3\theta)$.     * This graph is known as a "3 leaf rose."     * One cycle is completed on $[0, \pi)$.     * To find the interval of a single small loop, solve for $r = 0$:         * $0 = \cos(3\theta)$. Substitute $u = 3\theta$.         * $u = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}$.         * $\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}$.     * The interval \frac{\pi}{6} \le \theta < \frac{\pi}{2} describes one loop. At $\theta = \frac{\pi}{3}$, $r = \cos(\pi) = -1$, putting this loop in the third quadrant.

Converting Equations Between Systems

• Example 10 (Cartesian to Polar): $x^2 + y^2 = 6y$.     * Substitute $x^2 + y^2 = r^2$ and $y = r \times \sin(\theta)$.     * $r^2 = 6 \times r \times \sin(\theta)$.     * $r^2 - 6 \times r \times \sin(\theta) = 0 \implies r(r - 6 \times \sin(\theta)) = 0$.     * Excluding the point $r = 0$, the equation is $r = 6 \times \sin(\theta)$. This is a circle.
• Example 11 (Linear Equation): $y = 3x + 2$.     * Substitute $r \times \sin(\theta) = 3 \times r \times \cos(\theta) + 2$.     * $r \times \sin(\theta) - 3 \times r \times \cos(\theta) = 2$.     * $r(\sin(\theta) - 3 \times \cos(\theta)) = 2 \implies r = \frac{2}{\sin(\theta) - 3 \times \cos(\theta)}$.
• Example 12 (Polar to Cartesian): $r = \frac{3}{1 - 2 \times \cos(\theta)}$.     * Clear the fraction: $r(1 - 2 \times \cos(\theta)) = 3$.     * Use $\cos(\theta) = \frac{x}{r}$: $r - 2x = 3 \implies r = 3 + 2x$.     * Square both sides: $r^2 = (3 + 2x)^2$.     * Substitute $r^2 = x^2 + y^2$: $x^2 + y^2 = (3 + 2x)^2$.
• Example 13 (Double Angle Identification): $r = \sin(2\theta)$.     * Use double angle identity: $r = 2 \times \sin(\theta) \times \cos(\theta)$.     * Substitute $\sin(\theta) = \frac{y}{r}$ and $\cos(\theta) = \frac{x}{r}$: $r = 2 \times (\frac{y}{r}) \times (\frac{x}{r})$.     * Simplify: $r^3 = 2xy$.     * Since $r^2 = x^2 + y^2$, then $(x^2 + y^2)^{3/2} = 2xy$ or $(x^2 + y^2)^3 = (2xy)^2$.

Collection of Try It Now Questions and Answers

• Try It Now 1: Plot and label given coordinates:     * A: $(3, \frac{\pi}{6})$     * B: $(-2, \frac{\pi}{3})$     * C: $(3, \frac{4 heta}{4})$ [Note: likely $\frac{3 heta}{4}$ or similar according to symbols].     * Answers provided graphically in textbook.
• Try It Now 2:     * a. Convert $(r, \theta) = (2, \pi)$ to $(x, y)$. Result: $(-2, 0)$.     * b. Convert $(x, y) = (0, -4)$ to $(r, \theta)$. Result: $(4, \frac{3\pi}{2})$.
• Try It Now 3: Sketch $r = 3 \times \sin(\theta)$ and find the cycle interval.     * Returns to origin at $\theta = 0, \pi$. Interval: 0 \le \theta < \pi.
• Try It Now 4: Sketch $r = \sin(2\theta)$.     * This function is a 4-leaf rose.
• Try It Now 5:     * a. Rewrite $x^2 + y^2 = 3$ in polar form. Result: $r = \sqrt{3}$.     * b. Rewrite $r = 2 \times \sin(\theta)$ in Cartesian form. Result: $x^2 + y^2 = 2y$.

Summary of Section 8.2 Exercises

• Point Conversions: Exercise sets 1–12 emphasize polar to Cartesian; 13–20 emphasize Cartesian to polar (including non-integer coordinates like $(3, 2)$).
• Equation Conversions: Exercise sets 21–28 cover Cartesian to polar; 29–36 cover polar to Cartesian (including secant and cosecant forms).
• Matching and Graphing: Exercises 37–48 involve matching equations to specific graph shapes (circles, lima%E7ons, roses, and logarithmic spirals).
• Named Curves: Specific complex curves listed include:     * Conchoid: $r = 3 + \sec(\theta)$     * Lituus: $r^2 = \frac{1}{\theta}$     * Cissoid: $r = 2 \times \sin(\theta) \times \tan(\theta)$     * Hippopede: $r^2 = 2(1 - 2 \times \sin^2(\theta))$ [Approximate form for $r^2 = 2(1 - \sin(\theta))$].