Welcome to Coordinate Systems: Navigating the Mathematical Plane!

Hello future mathematician! This chapter is your guide to navigating the mathematical world using different maps. We are moving beyond the familiar grid system (Cartesian coordinates) and introducing a powerful alternative: Polar Coordinates.

In Further Pure Mathematics 1 (FP1), understanding how to switch smoothly between these systems is crucial. It allows us to define and analyze complex curves and shapes much more simply. Don't worry if this seems tricky at first—we'll break down the conversions step-by-step!

Part 1: Reviewing Cartesian Coordinates (The Familiar System)

The system you already know is the Cartesian system (or rectangular coordinates), defined by René Descartes. It uses two perpendicular axes, \(x\) and \(y\).

  • A point is defined by \((x, y)\).
  • \(x\) is the horizontal distance from the origin.
  • \(y\) is the vertical distance from the origin.

Think of this as giving directions in a city: go 3 blocks East (x=3), then 4 blocks North (y=4).

Key Takeaway

The Cartesian system relies on right-angle movement (horizontal and vertical).

Part 2: Introducing Polar Coordinates (A New Perspective)

The Polar Coordinate System defines a position based not on a grid, but on distance and direction relative to the origin.

What are Polar Coordinates?

A point \(P\) is defined by the coordinates \((r, \theta)\):

  • \(r\) (The Radial Distance): This is the straight-line distance from the origin (or pole) to the point \(P\). \(r\) is always non-negative (\(r \ge 0\)).
  • \(\theta\) (The Angle or Argument): This is the angle measured anticlockwise from the positive \(x\)-axis (the initial line) to the line segment \(OP\).

Important Note: In FP1, \(\theta\) is usually given in radians, and its range is typically specified, often \(0 \le \theta < 2\pi\) or \(-\pi < \theta \le \pi\).

Analogy: Think of a radar screen. A target is located by how far away it is (r) and its bearing or direction (\(\theta\)).

Plotting Points in Polar Form

To plot a point like \((4, \frac{\pi}{6})\):

  1. Start at the positive \(x\)-axis.
  2. Rotate anticlockwise by the angle \(\theta = \frac{\pi}{6}\) (30°).
  3. Move outwards along that line a distance of \(r = 4\).

Did you know? A single point in the plane can have infinitely many polar coordinates! For example, \((2, \frac{\pi}{2})\) is the same point as \((2, \frac{\pi}{2} + 2\pi)\) or \((2, \frac{5\pi}{2})\).

Key Takeaway

Polar coordinates use distance \(r\) and angle \(\theta\). They provide a powerful way to describe circular or rotational motion.

Part 3: Switching Between Systems (The Conversion Toolkit)

The beauty of coordinate geometry is that we can describe the same point using both systems. We use simple trigonometry (SOH CAH TOA) to switch between them.

Imagine a point \(P(x, y)\) forming a right-angled triangle with the origin and the \(x\)-axis. The hypotenuse is \(r\).

Conversion 1: Polar \((r, \theta)\) to Cartesian \((x, y)\)

This is the easiest conversion. You are given \(r\) and \(\theta\), and you need to find \(x\) and \(y\).

Formulas:

\[x = r \cos \theta\] \[y = r \sin \theta\]

Step-by-Step Example: Convert \((r, \theta) = (6, \frac{2\pi}{3})\) to Cartesian.

  1. Find \(x\): \(x = 6 \cos(\frac{2\pi}{3}) = 6 \times (-\frac{1}{2}) = -3\)
  2. Find \(y\): \(y = 6 \sin(\frac{2\pi}{3}) = 6 \times (\frac{\sqrt{3}}{2}) = 3\sqrt{3}\)
  3. The Cartesian point is \((-3, 3\sqrt{3})\).
Conversion 2: Cartesian \((x, y)\) to Polar \((r, \theta)\)

This is slightly trickier because we need to correctly determine the angle \(\theta\).

Step A: Finding \(r\)

We use the Pythagorean theorem:

Formula for \(r\):

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

(Since \(r\) is distance, we only take the positive root.)

Step B: Finding \(\theta\) (The Trickiest Part!)

We use the tangent function:

Formula for \(\theta\):

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

WARNING: Common Mistake to Avoid!

Using the inverse tangent function, \(\theta = \arctan(\frac{y}{x})\), only gives you an angle in Quadrant I or IV (between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)). You must check the quadrant of the original Cartesian point \((x, y)\) to adjust your angle!

Example: Convert \((x, y) = (-3, -3)\) to Polar.

  1. Find \(r\): \(r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\)
  2. Find reference angle (\(\alpha\)): Use the positive values: \(\alpha = \arctan(\frac{3}{3}) = \arctan(1) = \frac{\pi}{4}\).
  3. Check Quadrant: Since \(x\) is negative and \(y\) is negative, the point \((-3, -3)\) is in Quadrant III.
  4. Adjust \(\theta\): In Quadrant III, the angle is \(\pi + \alpha\). \[\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\]

The polar coordinate is \((3\sqrt{2}, \frac{5\pi}{4})\).

Quick Review: Quadrant Check for \(\theta\) (If \(0 \le \theta < 2\pi\))
  • QI (x > 0, y > 0): \(\theta = \alpha\)
  • QII (x < 0, y > 0): \(\theta = \pi - \alpha\)
  • QIII (x < 0, y < 0): \(\theta = \pi + \alpha\)
  • QIV (x > 0, y < 0): \(\theta = 2\pi - \alpha\)
Key Takeaway

Conversions rely on substituting \(x = r \cos \theta\) and \(y = r \sin \theta\). Always check the quadrant when finding \(\theta\).

Part 4: Equations of Curves in Polar Form

One of the main reasons we use polar coordinates in FP1 is to simplify the equation of curves, especially those that involve circles or rotational symmetry.

Toolbox for Conversion (Substitution Rules)

When converting an equation, you primarily use these four relationships:

  1. Substitute \(x = r \cos \theta\)
  2. Substitute \(y = r \sin \theta\)
  3. Substitute \(r^2 = x^2 + y^2\)
  4. Substitute \(\tan \theta = \frac{y}{x}\)
Simple Polar Loci (Shapes Defined by \(r\) or \(\theta\))

These are the simplest curves to define in the polar system:

1. \(r = a\) (where \(a\) is a constant)

This means the distance from the origin is always \(a\). This defines a circle centered at the origin with radius \(a\).

Conversion Example:

Cartesian equation: \(x^2 + y^2 = 25\)

Substitute \(r^2 = x^2 + y^2\): \(r^2 = 25\)

Polar equation: \(r = 5\)

2. \(\theta = \alpha\) (where \(\alpha\) is a constant angle)

This means the angle is fixed, but the distance \(r\) can be anything. This defines a straight line passing through the origin at an angle \(\alpha\).

Conversion Example:

Cartesian equation: \(y = x\) (This line is at 45°)

Substitute \(\tan \theta = y/x\): \(\tan \theta = x/x = 1\)

Polar equation: \(\theta = \frac{\pi}{4}\)

Converting Complex Equations (Step-by-Step)
Example A: Convert \(r = 2a \cos \theta\) (Polar) to Cartesian

This curve describes a circle centered not at the origin, but on the \(x\)-axis.

  1. We need \(r \cos \theta\) (which is \(x\)) and \(r^2\) (which is \(x^2 + y^2\)).
  2. Multiply both sides by \(r\): \[r^2 = 2ar \cos \theta\]
  3. Substitute the Cartesian relationships: \[x^2 + y^2 = 2ax\]
  4. (Optional: Complete the square to show it's a circle) \[x^2 - 2ax + y^2 = 0\] \[(x - a)^2 - a^2 + y^2 = 0\] \[(x - a)^2 + y^2 = a^2\] This is a circle with center \((a, 0)\) and radius \(a\).
Example B: Convert \(x = 4\) (Cartesian) to Polar

This is a vertical line.

  1. Substitute \(x = r \cos \theta\): \[r \cos \theta = 4\]
  2. Solve for \(r\) (the standard polar form is often \(r = f(\theta)\)): \[r = \frac{4}{\cos \theta}\] or \[r = 4 \sec \theta\]
Tip for Struggling Students

When converting from Cartesian to Polar, try to force the terms \(x^2 + y^2\) (which becomes \(r^2\)) and \(x^2 / y^2\) (which gives \(\tan \theta\)). When converting from Polar to Cartesian, try to multiply by \(r\) if you see terms like \(\cos \theta\) or \(\sin \theta\) alone, so you can create \(r \cos \theta\) or \(r \sin \theta\).

Key Takeaway

Conversions simplify the analysis of curves. Remember the fundamental substitutions: \(x=r\cos\theta\), \(y=r\sin\theta\), and \(r^2=x^2+y^2\).

Common Mistakes and Study Tips

To ensure success in this chapter, focus on these details:

1. Mistake: Ignoring the Quadrant.

NEVER rely solely on \(\arctan(y/x)\) when converting Cartesian to Polar. Always sketch the point \((x, y)\) first to determine the correct quadrant for \(\theta\).

2. Mistake: Confusing \(r\) and \(\theta\).

Remember the order is \((r, \theta)\). \(r\) is distance, \(\theta\) is angle (rotation). If the angle is \(\frac{\pi}{2}\), the point is on the positive \(y\)-axis, regardless of \(r\).

3. Study Tip: Practice the Simple Loci.

Make sure you instantly recognize that \(r=a\) is a circle and \(\theta=\alpha\) is a line. These are fundamental building blocks.

4. Memory Aid: The Conversion Triangle

Draw the right-angled triangle. Hypotenuse \(r\), horizontal leg \(x\), vertical leg \(y\). All conversion formulas follow directly from this triangle and SOH CAH TOA.

5. Units: Always assume angles are in radians unless degrees are explicitly stated.

You've mastered two ways of mapping the world! This dual perspective will be incredibly useful as you proceed through advanced mathematics. Keep practicing those conversions!