Polar Coordinates and Graphs

🧭 Polar Coordinates and Graphs

Part 1 of 7 — Locating a Point with $(r, \theta)$

Topics in This Part

🔑 Key Concept: In rectangular coordinates you reach a point by going right then up: $(x, y)$. In polar coordinates you reach it by pointing in a direction, then walking out: $(r, \theta)$. The radius $r$ is the distance from the origin (the pole); the angle $\theta$ is measured counterclockwise from the positive $x$-axis (the polar axis).

The Polar Plane

A point's polar coordinates $(r, \theta)$ answer two questions:

• $r$ — how far? the directed distance from the pole (origin).
• $\theta$ — which way? the angle, measured counterclockwise from the positive $x$-axis.

Plotting $(r, \theta)$ when $r > 0$

1. Face along the polar axis (positive $x$-direction).
2. Rotate counterclockwise by $\theta$.
3. Walk forward a distance $r$.

💡 Angles in AP Precalculus are usually in radians: $\dfrac{\pi}{2} = 90^\circ$, $\pi = 180^\circ$, $\dfrac{3\pi}{2} = 270^\circ$, $2\pi = 360^\circ$.

Concept Check 🎯

Match the Angle 🔽

Match each radian angle with the direction it points along the polar axis.

Reading the Grid

A polar grid is a set of concentric circles (constant $r$) crossed by rays from the pole (constant $\theta$).

• A point on the circle $r = 2$ is exactly $2$ units from the pole, somewhere.
• A point on the ray $\theta = \dfrac{\pi}{3}$ lies along that $60^\circ$ direction, at some distance.
• Their intersection pins down the single point $\left(2, \dfrac{\pi}{3}\right)$.

🔑 Key Idea: "Circle picks the distance, ray picks the direction." Specifying both $r$ and $\theta$ identifies a point — though, as Part 2 shows, that point has many polar names.

Concept Check 🎯

🧭 Polar Coordinates and Graphs

Part 2 of 7 — Non-Uniqueness & Negative $r$

🔑 The Twist: Unlike rectangular coordinates, one point has infinitely many polar names. Adding $2\pi$ to $\theta$ returns to the same direction, and a negative radius points the opposite way.

Same Point, Many Names

1. Add full turns to $\theta$

Rotating an extra $2\pi$ (or any multiple) lands in the same direction:

$\left(2, \dfrac{\pi}{6}\right) = \left(2, \dfrac{\pi}{6} + 2\pi\right) = \left(2, \dfrac{13\pi}{6}\right)$

2. Negative radius: walk backward

A negative $r$ means: face direction $\theta$, then walk backward. Equivalently, flip the direction by $\pi$:

$(-r,\, \theta) = (r,\, \theta + \pi)$

Example: $\left(-2, \dfrac{\pi}{6}\right)$ faces $30^\circ$ but walks backward, ending up the same as $\left(2, \dfrac{\pi}{6} + \pi\right) = \left(2, \dfrac{7\pi}{6}\right)$.

⚠️ Watch out: $(-r, \theta)$ is not the reflection of $(r,\theta)$ across the $x$-axis. It is the point directly opposite through the pole — add $\pi$ to the angle.

Concept Check 🎯

The Pole Itself

The pole (origin) is special: $r = 0$ regardless of the angle.

$(0, \theta) = (0, 0) = (0, \pi) = (0, \text{anything})$

When you have $r = 0$ at some angle, the graph passes through the pole there — a fact we'll use to find where polar curves touch the origin.

💡 Convention for naming a point: AP problems often ask for the name with $r > 0$ and $0 \le \theta < 2\pi$. To get there: if $r < 0$, replace $(r, \theta)$ with $(|r|, \theta + \pi)$; then add/subtract $2\pi$ until $\theta$ is in $[0, 2\pi)$.

Standardize the Name 🔽

Rewrite each point with $r > 0$ and $0 \le \theta < 2\pi$.

Concept Check 🎯

🧭 Polar Coordinates and Graphs

Part 3 of 7 — Converting Between Polar and Rectangular

🔑 The Bridge: A right triangle with hypotenuse $r$ and angle $\theta$ connects the two systems. Memorize these four relationships and you can move freely between $(r,\theta)$ and $(x,y)$.

The Conversion Formulas

Drop a perpendicular from the point to the $x$-axis. The legs are $x$ and $y$, the hypotenuse is $r$, and the angle at the origin is $\theta$.

Polar → Rectangular

$x = r\cos\theta \qquad y = r\sin\theta$

Rectangular → Polar

$r = \sqrt{x^2 + y^2} \qquad \tan\theta = \dfrac{y}{x}$

⚠️ Quadrant check: $\tan\theta = \dfrac{y}{x}$ alone can't tell quadrant II from IV (both give the same ratio sign issues). Always look at the signs of $x$ and $y$ and place $\theta$ in the correct quadrant.

Worked Example: $\left(4, \dfrac{\pi}{3}\right) \to$ rectangular

$x = 4\cos\dfrac{\pi}{3} = 4\cdot\dfrac{1}{2} = 2$ $y = 4\sin\dfrac{\pi}{3} = 4\cdot\dfrac{\sqrt3}{2} = 2\sqrt3$

So $\left(4, \dfrac{\pi}{3}\right) = \left(2,\, 2\sqrt3\right)$.

Polar → Rectangular 🧮

Convert each point. Use $x = r\cos\theta$, $y = r\sin\theta$. Enter exact values (e.g. -2, 0, or a decimal).

1) $\left(6, \dfrac{\pi}{2}\right)$: $x = \,?$ 2) Same point: $y = \,?$ 3) $\left(2, \pi\right)$: $x = \,?$

Rectangular → Polar: Worked Example

Convert $(-1, \sqrt3)$ to polar with $r > 0$, $0 \le \theta < 2\pi$.

Step 1 — find $r$: $r = \sqrt{(-1)^2 + (\sqrt3)^2} = \sqrt{1 + 3} = \sqrt4 = 2$

Step 2 — find the reference angle: $\tan\theta = \dfrac{y}{x} = \dfrac{\sqrt3}{-1} = -\sqrt3 \;\Rightarrow\; \text{reference } \dfrac{\pi}{3}$

Step 3 — pick the quadrant: $x < 0$, $y > 0$ ⟹ Quadrant II, so $\theta = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$

Answer: $(-1, \sqrt3) = \left(2, \dfrac{2\pi}{3}\right)$.

💡 The calculator's $\arctan$ only returns angles in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$. Use the quadrant of $(x,y)$ to add $\pi$ (Q II / Q III) when needed.

Concept Check 🎯

Rectangular → Polar 🧮

Find $r$ (with $r > 0$) for each point. Enter exact values; use sqrt style decimals are fine (e.g. 2.83) but exact integers are preferred.

1) $(3, 4)$: $r = \,?$ 2) $(0, 7)$: $r = \,?$ 3) $(-5, 12)$: $r = \,?$

🧭 Polar Coordinates and Graphs

Part 4 of 7 — Graphing $r = f(\theta)$ Point by Point

🔑 The Core Skill: A polar function $r = f(\theta)$ assigns a radius to each angle. To graph it, sweep $\theta$ around, compute $r$ at each angle, plot the points, and connect them smoothly.

Building a Table

To graph $r = f(\theta)$:

1. Choose convenient angles ($0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \dfrac{\pi}{2}, \dots$).
2. Compute $r = f(\theta)$ for each.
3. Plot each $(r, \theta)$ — remember a negative $r$ reflects across the pole.
4. Connect in order of increasing $\theta$.

Worked Example: $r = 4\cos\theta$

This traces a circle of radius $2$ centered at $(2, 0)$. (At $\theta = \pi$, $r = -4$ lands back on the right side — the negatives retrace the same circle.)

💡 $r = a\cos\theta$ is a circle of diameter $a$ sitting on the $x$-axis; $r = a\sin\theta$ is a circle of diameter $a$ on the $y$-axis.

Fill the Table 🧮

For $r = 2 + 2\cos\theta$ (a cardioid), compute $r$ at each angle.

1) $\theta = 0$: $r = \,?$ 2) $\theta = \dfrac{\pi}{2}$: $r = \,?$ 3) $\theta = \pi$: $r = \,?$

Plotting Negative $r$

When $f(\theta) < 0$, plot the point opposite the angle (across the pole). This is exactly the rule $(-r, \theta) = (r, \theta + \pi)$.

Example: $r = 3\cos(2\theta)$ at $\theta = \dfrac{\pi}{2}$

$r = 3\cos\!\left(2\cdot\dfrac{\pi}{2}\right) = 3\cos(\pi) = -3$

Facing $\dfrac{\pi}{2}$ (up) with $r = -3$ plots the point down, at $\left(3, \dfrac{3\pi}{2}\right)$. Negative radii are how rose curves form petals on the "opposite" side.

⚠️ Don't skip negative $r$ values — they are real points on the graph, just plotted in the opposite direction.

Concept Check 🎯

Identify the Circle 🔽

Match each polar equation to its circle.

🧭 Polar Coordinates and Graphs

Part 5 of 7 — The Curve Library: Roses, Cardioids & Limaçons

🔑 Pattern Recognition: AP Precalculus expects you to recognize a polar graph from its equation. This part is your field guide to the four families that show up again and again.

The Standard Families

Counting Rose Petals

For $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$:

• If $n$ is odd → the rose has $n$ petals.
• If $n$ is even → the rose has $2n$ petals.

💡 The petal length is $|a|$ — the maximum value of $|r|$.

Count the Petals 🧮

Use the odd/even rule. Enter the number of petals.

1) $r = 5\sin(3\theta)$: petals $= \,?$ 2) $r = 2\cos(4\theta)$: petals $= \,?$ 3) $r = \sin(6\theta)$: petals $= \,?$

Cardioids vs. Limaçons

Both come from $r = a \pm b\cos\theta$ (or $\sin$). The relationship between $a$ and $b$ decides the shape:

State the dimple rule cleanly using the ratio $\dfrac{a}{b}$:

• $\dfrac{a}{b} = 1$ → cardioid
• $\dfrac{a}{b} < 1$ → inner loop
• $1 < \dfrac{a}{b} < 2$ → dimpled
• $\dfrac{a}{b} \ge 2$ → convex (no dimple)

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

Here $a = 2$, $b = 3$, so $\dfrac{a}{b} = \dfrac{2}{3} < 1$ → limaçon with an inner loop. The loop appears where $r$ goes negative.

⚠️ A cardioid is the special limaçon with $a = b$. Every cardioid is a limaçon, but not every limaçon is a cardioid.

Classify the Curve 🔽

Use the ratio $\dfrac{a}{b}$ (for $r = a \pm b\cos\theta$ or $\sin$).

Concept Check 🎯

🧭 Polar Coordinates and Graphs

Part 6 of 7 — Rates of Change, Symmetry & Distance from the Pole

🔑 AP Focus (Skill 3.15): Beyond drawing the curve, AP Precalculus asks you to reason about $r$ as $\theta$ changes: where is the graph getting farther from or closer to the pole, where is it maximally far, and what does the average rate of change of $r$ tell you?

Increasing / Decreasing Distance from the Pole

For a polar function $r = f(\theta)$, the value $|r|$ is the distance from the pole.

• As $\theta$ increases, if $r$ is increasing (and positive), the curve spirals outward — farther from the pole.
• If $r$ is decreasing (and positive), the curve moves inward — closer to the pole.

Example: $r = 1 + 2\cos\theta$ on $\left[0, \pi\right]$

From $\theta = 0$ to $\theta = \dfrac{\pi}{2}$, $r$ drops from $3$ to $1$: the point is moving toward the pole. Maximum distance $|r| = 3$ occurs at $\theta = 0$.

💡 Maximum distance from the pole happens where $|r|$ is largest. For $r = a + b\cos\theta$ with $a, b > 0$, that's at $\theta = 0$ (cosine $= 1$), giving $r = a + b$.

Average Rate of Change of $r$ 🧮

The average rate of change of $r$ over $[\theta_1, \theta_2]$ is $\dfrac{f(\theta_2) - f(\theta_1)}{\theta_2 - \theta_1}$.

For $r = 1 + 2\cos\theta$ (from the table above):

1) $r$ at $\theta = 0$ is $\,?$ 2) $r$ at $\theta = \dfrac{\pi}{2}$ is $\,?$ 3) Over $\left[0, \dfrac{\pi}{2}\right]$, is the average rate of change positive or negative? Enter positive or negative.

Symmetry Tests

Symmetry cuts your plotting work in half. Test a polar equation by substituting:

Example: $r = 4\cos\theta$

Replace $\theta \to -\theta$: $r = 4\cos(-\theta) = 4\cos\theta$ (cosine is even) — unchanged, so it's symmetric about the polar axis. ✓

Example: $r = 4\sin\theta$

Replace $\theta \to \pi - \theta$: $r = 4\sin(\pi - \theta) = 4\sin\theta$ — unchanged, so it's symmetric about the line $\theta = \dfrac{\pi}{2}$. ✓

⚠️ These are sufficient tests, not necessary ones: a graph can be symmetric even if a test fails (because of polar non-uniqueness). A passing test guarantees symmetry; a failing test is inconclusive.

Concept Check 🎯

Through the Pole 🔽

A polar curve passes through the pole at any $\theta$ where $r = 0$. Find those angles on $[0, 2\pi)$.

🧭 Polar Coordinates and Graphs

Part 7 of 7 — Mixed Practice & Mastery Check

You can now (1) plot and name points in $(r, \theta)$, (2) handle negative $r$ and non-uniqueness, (3) convert both directions, (4) graph $r = f(\theta)$, (5) recognize the curve library, and (6) reason about rates, symmetry, and distance from the pole. Let's put it all together.

Quick Reference

⚠️ Remember the quadrant check on rectangular → polar, and don't drop negative $r$ values when graphing.

Mixed Practice 🎯

One More Conversion 🧮

Convert $\left(8, \dfrac{\pi}{6}\right)$ to rectangular coordinates. (Recall $\cos\frac{\pi}{6} = \frac{\sqrt3}{2}$, $\sin\frac{\pi}{6} = \frac{1}{2}$.)

1) $x = \,?$ (enter as a decimal, e.g. 6.93, or 4*sqrt3 is the exact form) 2) $y = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.