Pure Mathematics | Polar coordinates

Welcome to Polar Coordinates!

Hello future mathematicians! In FP2, we move beyond the familiar \( (x, y) \) Cartesian grid and explore a new, often much simpler, way to describe location: Polar Coordinates.

Don't worry if this sounds intimidating! Think of it this way: the standard Cartesian system is great for rectangles and straight lines, but if you want to describe a circle, a spiral, or a heart shape, using distances and angles is far more natural.

In this chapter, we will learn how to:

• Convert effortlessly between Cartesian and Polar systems.
• Sketch complex curves using polar equations.
• Calculate the area enclosed by these curves using integration.

Let's dive in!

Section 1: Defining Polar Coordinates \((r, \theta)\)

1.1 What are Polar Coordinates?

A point P in the Cartesian system is defined by its horizontal and vertical distances \( (x, y) \). In the polar system, a point P is defined by its distance from the origin and the angle it makes with the positive x-axis. We write this as \( (r, \theta) \).

The Key Ingredients:

• \(r\) (The Radius/Modulus): This is the straight-line distance from the origin (which we call the Pole) to the point P. \(r\) is usually positive, \(r \ge 0\).
• \(\theta\) (The Angle/Argument): This is the angle measured counter-clockwise from the positive x-axis (which we call the Initial Line) to the line segment OP. \(\theta\) is always measured in radians.

Analogy Alert! The Ship Navigator

Imagine you are navigating a ship from port (the Pole).
Instead of saying "Go 3 miles East and 4 miles North" (Cartesian), you say:
"Travel 5 miles out (\(r\)) at a bearing of 53 degrees (\(\theta\))."
Polar coordinates simplify descriptions of circular motion or paths radiating from a central point!

Key Takeaway: Polar coordinates define position using distance from the centre (r) and rotation from the x-axis (\(\theta\)).

Section 2: Converting Between Systems

Since both systems describe the same point P, we must be able to switch between them. This is done using basic trigonometry on the right-angled triangle formed by the point P, the origin, and the projection onto the x-axis.

2.1 Polar \((r, \theta)\) to Cartesian \((x, y)\)

This is the easiest conversion. If you know \(r\) and \(\theta\), SOH CAH TOA gives us \(x\) and \(y\) directly:

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

Example: The point \((4, \frac{\pi}{3})\) in polar coordinates converts to: $$ x = 4 \cos \left(\frac{\pi}{3}\right) = 4 \times \frac{1}{2} = 2 $$ $$ y = 4 \sin \left(\frac{\pi}{3}\right) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} $$ So the Cartesian point is \((2, 2\sqrt{3})\).

2.2 Cartesian \((x, y)\) to Polar \((r, \theta)\)

To find \(r\) and \(\theta\) from \(x\) and \(y\), we use Pythagoras and the tangent function.

Finding \(r\) (The Radius)

Use the Pythagorean Theorem: $$ r^2 = x^2 + y^2 \quad \text{or} \quad r = \sqrt{x^2 + y^2} $$ Since \(r\) is a distance, we usually take the positive root.

Finding \(\theta\) (The Angle)

Use the tangent function: $$ \tan \theta = \frac{y}{x} $$ $$ \theta = \arctan \left(\frac{y}{x}\right) $$

!!! Critical Warning: Choosing the Correct Quadrant for \(\theta\) !!!

This is the most common mistake students make. The calculator only gives an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). You must look at the signs of \(x\) and \(y\) to determine the correct quadrant:

• Q1 (\(x>0, y>0\)): \(\theta\) is the calculated angle.
• Q2 (\(x<0, y>0\)): \(\theta = \text{calculated angle} + \pi\).
• Q3 (\(x<0, y<0\)): \(\theta = \text{calculated angle} + \pi\).
• Q4 (\(x>0, y<0\)): \(\theta = \text{calculated angle} + 2\pi\) (or just use the negative angle from the calculator).

Always sketch the point \((x, y)\) first to see which quadrant you are in!

Section 3: Sketching Polar Curves \(r = f(\theta)\)

The real fun begins when we look at equations where \(r\) depends on \(\theta\). These equations, \(r = f(\theta)\), generate wonderful shapes!

3.1 General Procedure for Sketching

When you are asked to sketch a polar curve (usually over the range \(0 \le \theta \le 2\pi\)), follow these steps:

1. Check for Symmetry: This saves a lot of work!
   • Symmetry about the Initial Line (\(\theta = 0\), the x-axis): If replacing \(\theta\) with \(-\theta\) results in the same equation, the curve is symmetric about the x-axis. (e.g., \(r = a(1 + \cos \theta)\))
   • Symmetry about the Line \(\theta = \frac{\pi}{2}\) (the y-axis): If replacing \(\theta\) with \(\pi - \theta\) results in the same equation, the curve is symmetric about the y-axis. (e.g., \(r = a \sin 2\theta\))
2. Find Key Points: Calculate \(r\) for simple values of \(\theta\).
   • \(\theta = 0\): Where does the curve cross the positive x-axis?
   • \(\theta = \frac{\pi}{2}\): Where does it cross the positive y-axis?
   • \(\theta = \pi\): Where does it cross the negative x-axis?
   • \(\theta = \frac{3\pi}{2}\): Where does it cross the negative y-axis?
3. Check if it Passes through the Pole (\(r=0\)): Find the angles \(\theta\) for which \(r=0\). These are crucial as the curve loops back to the origin at these angles.
4. Use Intermediate Points: Calculate a few more points (e.g., \(\frac{\pi}{6}, \frac{\pi}{4}\)) to understand the curvature.

3.2 Common FP2 Polar Curves

You should be familiar with the shapes produced by these common equations:

1. Circles

• \(r = a\): A circle of radius \(a\) centred at the pole. (The simplest curve!)
• \(r = a \cos \theta\): A circle of diameter \(a\) passing through the pole, tangent to the y-axis, centred on the x-axis.
• \(r = a \sin \theta\): A circle of diameter \(a\) passing through the pole, tangent to the x-axis, centred on the y-axis.

2. Cardioids (Heart Shapes)

• \(r = a(1 + \cos \theta)\) or \(r = a(1 - \cos \theta)\): Symmetric about the x-axis. These curves always pass through the pole at some angle.
• \(r = a(1 + \sin \theta)\) or \(r = a(1 - \sin \theta)\): Symmetric about the y-axis.

Did you know? The word 'Cardioid' comes from the Greek word 'kardia' meaning heart.

3. Limacons
Limacons have the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). The shape depends entirely on the ratio of \(a\) and \(b\).

• If \(|a| = |b|\): It’s a Cardioid (we already covered this).
• If \(|a| > |b|\): It’s a Dimpled Limacon (no inner loop, never reaches the pole).
• If \(|a| < |b|\): It’s a Limacon with an Inner Loop (it crosses the pole twice).

4. Spirals

• \(r = a \theta\): The distance \(r\) continuously increases as the angle \(\theta\) increases, forming a spiral that winds outwards from the pole.

Struggling Students Tip: If sketching is tough, focus on the algebra first. If \(r = 2 \cos \theta\), know that when \(\theta = 0\), \(r=2\). When \(\theta = \pi/2\), \(r=0\). This tells you the curve starts at (2, 0) and immediately returns to the origin!

Key Takeaway: Sketching involves testing symmetry, finding intercepts, and determining angles where \(r=0\) (the pole). The shape depends heavily on whether \(r\) is a function of \(\cos \theta\) or \(\sin \theta\).

Section 4: Area Enclosed by a Polar Curve

One of the primary applications of polar coordinates in FP2 is calculating the area swept out by a curve. In Cartesian coordinates, we used thin vertical rectangles to approximate area (\(\int y \, dx\)). In polar coordinates, we use thin triangular sectors (like tiny slices of pizza!).

4.1 The Area Formula

The area \(A\) bounded by the curve \(r = f(\theta)\) and the radial lines \(\theta = \alpha\) and \(\theta = \beta\) is given by the definite integral:

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

Prerequisite check: Remember the area of a sector in radians is \(\frac{1}{2} r^2 \theta\). The integral is just summing up infinitely thin sectors where \(\theta\) becomes \(d\theta\).

4.2 Steps for Calculating Area

Step 1: Identify the limits of integration, \(\alpha\) and \(\beta\).

• If the question asks for the area between two specific lines, those lines define \(\alpha\) and \(\beta\).
• If the question asks for the total area of a closed loop (like a cardioid), you must find the angles that trace the loop exactly once (often from \(\alpha = 0\) to \(\beta = 2\pi\), or sometimes just \(0\) to \(\pi\) and double the result if there is symmetry).

Step 2: Substitute \(r\) into the integral.
Since \(r\) is usually \(f(\theta)\), you must calculate \((f(\theta))^2\).

Example: If \(r = a(1 + \cos \theta)\), then \(r^2 = a^2 (1 + \cos \theta)^2\).

Step 3: Simplify and Integrate.
This step requires strong trigonometric integration skills, specifically the Double Angle Identities, because you will often end up needing to integrate \(\cos^2 \theta\) or \(\sin^2 \theta\).

Memory Aid for Integration:

• $$ \cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta) $$
• $$ \sin^2 \theta = \frac{1}{2}(1 - \cos 2\theta) $$

You must convert squared trig functions into terms involving \(\cos 2\theta\) before integration.

Step 4: Evaluate the definite integral.
Substitute the limits \(\beta\) and \(\alpha\). Remember that the integral of \(\cos k\theta\) is \(\frac{1}{k} \sin k\theta\).

4.3 Area Between Two Polar Curves

If you need the area between two curves, \(r_1 = f_1(\theta)\) (the outer curve) and \(r_2 = f_2(\theta)\) (the inner curve), the area is the difference between the areas they enclose:

$$ A = \frac{1}{2} \int_{\alpha}^{\beta} (r_1^2 - r_2^2) \, d\theta $$

You must clearly identify which radius is the outer one (\(r_1\)) in the region you are integrating over! Sketching is essential here.

Key Takeaway: The area formula is \( A = \frac{1}{2} \int r^2 \, d\theta \). Mastering the substitution \(r^2\) and subsequent trig integration is the key to success here.

Summary and Final Encouragement

You have now mastered the fundamentals of polar coordinates! This topic links together geometry, trigonometry, and integration beautifully. While the conversion and sketching require careful attention to quadrants and symmetry, the integration is highly mechanical once you remember the key identity (\(\cos^2 \theta\)).

Practice those common curve sketches and integration techniques, and you will find that polar coordinates are a powerful tool in your FP2 toolkit. Keep practicing—you've got this!