Pure Mathematics | Further coordinate systems

Welcome to Further Coordinate Systems!

Hello mathematicians! You’ve already mastered the classic \(x-y\) (Cartesian) coordinate system. That system is brilliant for straight lines and rectangles, but sometimes, it makes complex curves look incredibly difficult. That’s where this exciting chapter comes in!

In "Further Coordinate Systems," we're diving into the world of Polar Coordinates. This system is essential for describing motion, rotations, and shapes that are naturally circular or spiral. Mastering this topic will give you powerful new tools for calculus and problem-solving, opening the door to beautifully symmetrical curves that are tough to handle otherwise.

Don't worry if this seems tricky at first—we’ll break down every concept step-by-step. Let’s get started!

Section 1: Cartesian Coordinates – A Quick Review

Before we introduce the new system, let’s quickly remind ourselves of the system we already know: the Cartesian System, where a point \(P\) is defined by \((x, y)\).

• \(x\) is the horizontal distance from the origin.
• \(y\) is the vertical distance from the origin.

This system uses perpendicular axes (the x-axis and the y-axis).

Prerequisite Check:

You must be confident with basic trigonometry (SOH CAH TOA) and the Pythagorean theorem, as we use them extensively to switch between systems.

Section 2: Introduction to Polar Coordinates

Imagine you are working in air traffic control or marine navigation. Instead of asking "how far East and how far North is the target?", it’s much easier to ask "how far away is the target, and in what direction?".

This is the fundamental idea behind Polar Coordinates.

Defining Polar Coordinates \((r, \theta)\)

A point \(P\) in the plane is defined by two values: \((r, \theta)\).

1. \(r\) (The Radius)

• This is the direct distance from the origin (called the pole) to the point \(P\).
• It is conventionally defined such that \(r \ge 0\).

2. \(\theta\) (The Angle)

• This is the angle, measured anti-clockwise, from the initial line (the positive x-axis) to the line segment \(OP\).
• \(\theta\) is usually measured in radians, and typically constrained to \(0 \le \theta < 2\pi\) or \(-\pi < \theta \le \pi\).

Key Terminology:

• The Origin \((0, 0)\) in Cartesian is called the Pole.
• The positive x-axis is called the Initial Line.

Analogy: The Searchlight

Think of the origin as a searchlight. To locate a point, you first rotate the light by the angle \(\theta\), and then you extend the beam by the distance \(r\). This is often much simpler for describing circular movements than using horizontal and vertical movements.

Polar Coordinates and Negative \(r\) (Advanced Note)

While standard conventions often require \(r \ge 0\), you might encounter problems where \(r\) can be negative, especially when sketching complex curves. If \(r\) is negative, it means you move \(|r|\) units in the direction opposite to \(\theta + \pi\). Always check the specific definition required by the question!

Section 3: Converting Between Cartesian and Polar Coordinates

The ability to switch fluently between the two systems is crucial.

A) Polar to Cartesian: \((r, \theta) \rightarrow (x, y)\)

Look at the right-angled triangle formed by the pole, the point \((r, \theta)\), and the projection onto the initial line. Using SOH CAH TOA:

The formulas are:

$$x = r \cos \theta$$

$$y = r \sin \theta$$

Example: Convert \((r, \theta) = (4, \frac{\pi}{6})\) to Cartesian.

• \(x = 4 \cos(\frac{\pi}{6}) = 4 (\frac{\sqrt{3}}{2}) = 2\sqrt{3}\)
• \(y = 4 \sin(\frac{\pi}{6}) = 4 (\frac{1}{2}) = 2\)
• Cartesian coordinates: \((2\sqrt{3}, 2)\)

Memory Aid: Remember the order of \((x, y)\) follows the order of trigonometric functions alphabetically: Cosine for x, Sine for y.

B) Cartesian to Polar: \((x, y) \rightarrow (r, \theta)\)

We use Pythagoras and inverse trigonometry.

1. Finding \(r\) (Distance)

$$r^2 = x^2 + y^2$$

$$r = \sqrt{x^2 + y^2}$$

2. Finding \(\theta\) (Angle)

We know that \(\tan \theta = \frac{y}{x}\). Therefore:

$$\theta = \arctan \left(\frac{y}{x}\right)$$

!!! CRITICAL STEP: Quadrant Check !!!

The \(\arctan\) function only gives angles in Quadrants I and IV (between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)). You must adjust the angle based on the quadrant of \((x, y)\):

• Q I (\(x > 0, y > 0\)): \(\theta\) is the calculated angle.
• Q II (\(x < 0, y > 0\)): \(\theta = \text{calculated angle} + \pi\).
• Q III (\(x < 0, y < 0\)): \(\theta = \text{calculated angle} + \pi\) (or \(\theta = \text{calculated angle} - \pi\) if you prefer the range \(-\pi < \theta \le \pi\)).
• Q IV (\(x > 0, y < 0\)): \(\theta\) is the calculated angle (negative angle), or \(\theta = 2\pi + \text{calculated angle}\).

Common Mistake to Avoid: Not adjusting \(\theta\) for Quadrants II and III. Always sketch the point \((x, y)\) first!

C) Converting Equations

Sometimes you need to convert an entire equation, not just a single point.

1. Cartesian Equation to Polar Form: Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\). Also use \(x^2 + y^2 = r^2\).

Example: Convert the circle \(x^2 + y^2 = 9\).

Since \(x^2 + y^2 = r^2\), the polar form is \(r^2 = 9\), or simply \(r = 3\). (Much simpler!)

2. Polar Equation to Cartesian Form: Substitute \(r = \sqrt{x^2 + y^2}\), \(\cos \theta = \frac{x}{r}\), and \(\sin \theta = \frac{y}{r}\).

Example: Convert \(r = 2 \cos \theta\).

Multiply both sides by \(r\): \(r^2 = 2r \cos \theta\).
Substitute: \(x^2 + y^2 = 2x\).
This is the Cartesian equation of a circle: \((x-1)^2 + y^2 = 1\).

Section 4: Sketching Curves in Polar Coordinates

Sketching polar curves involves choosing key values of \(\theta\), calculating the corresponding \(r\), and then plotting these points.

Step-by-Step Sketching Strategy

1. Identify Symmetry: Check if the curve has symmetry.
   • If the equation is unchanged when \(\theta\) is replaced by \(-\theta\), the curve is symmetric about the initial line (x-axis).
   • If the equation is unchanged when \(\theta\) is replaced by \(\pi - \theta\), the curve is symmetric about the y-axis (\(\theta = \pi/2\)).
2. Determine the Range: Find the range of \(\theta\) needed to sketch the entire curve (often \(0\) to \(2\pi\), but sometimes less, like \(0\) to \(\pi\) for circles).
3. Plot Key Points: Calculate \(r\) for simple values of \(\theta\) (e.g., \(0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\), etc.).
4. Handle \(r=0\): Find the values of \(\theta\) for which \(r=0\). These points are often where the curve passes through the pole (the origin).
5. Identify Maximum \(r\): Find the largest value of \(r\). This is the point farthest from the origin.

Common Polar Curve Families

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

1. Circles (The Simplest)

• \(r = a\): A circle, radius \(a\), centered at the pole.
• \(r = 2a \cos \theta\): A circle, diameter \(2a\), passing through the pole, centered on the initial line.
• \(r = 2a \sin \theta\): A circle, diameter \(2a\), passing through the pole, centered on the line \(\theta = \pi/2\).

2. Cardioids (Heart Shapes) and Limacons

Equations of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\).

• Cardioid (\(a = b\)): \(r = a(1 + \cos \theta)\). Passes through the pole and has a 'cusp' (a sharp point) there. It looks like a heart.
• Limacon with inner loop (\(a < b\)): \(r = a + b \cos \theta\). Loops through the origin twice.

3. Roses (Petal Curves)

Equations of the form \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\).

• If \(n\) is odd, the rose has \(n\) petals. (Sketch range is \(0 \le \theta < \pi\))
• If \(n\) is even, the rose has \(2n\) petals. (Sketch range is \(0 \le \theta < 2\pi\))

Did you know?

The spirals of nautilus shells and the path of planets orbiting a star are often best described using generalized polar or spherical coordinate systems, demonstrating their importance in physics and nature!

Section 5: Calculating Area in Polar Coordinates

Finding the area enclosed by a polar curve is one of the primary calculus applications of this system. We cannot use the standard Cartesian integration formula \(\int y \, dx\).

The Area Formula

In Cartesian coordinates, area is approximated by rectangles. In polar coordinates, we approximate the area using sectors of circles.

Recall that the area of a circular sector with radius \(r\) and angle \(d\theta\) is approximately \(\frac{1}{2} r^2 d\theta\).

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

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

(Note: \(r\) is replaced by \(f(\theta)\) in the integrand.)

Step-by-Step Area Calculation

1. Determine Limits (\(\alpha\) and \(\beta\)):

• If the question asks for the area of a specific section, \(\alpha\) and \(\beta\) are given.
• If the question asks for the total area enclosed by the curve (e.g., a Cardioid), you must find the limits where the curve starts and ends, often where \(r=0\).

2. Set up the Integral: Substitute the expression for \(r\) into the formula, remembering to square it!

3. Integrate: This usually requires using trigonometric identities, particularly the Double Angle Formula for Cosine:

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

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

Integrating \(r^2\) often involves terms like \(\sin^2\theta\) or \(\cos^2\theta\), which must be converted before integration.

Example Walkthrough: Area of a Cardioid

Find the total area enclosed by \(r = a(1 + \cos \theta)\).

1. Symmetry: The curve is symmetric about the initial line. We can calculate the area for \(0 \le \theta \le \pi\) and double the result.
2. Limits: For the upper half, we sweep from \(\theta = 0\) to \(\theta = \pi\).
3. Setup: $$A_{\text{half}} = \frac{1}{2} \int_{0}^{\pi} [a(1 + \cos \theta)]^2 \, d\theta$$ $$A_{\text{half}} = \frac{a^2}{2} \int_{0}^{\pi} (1 + 2\cos \theta + \cos^2 \theta) \, d\theta$$
4. Substitute Identity: Replace \(\cos^2 \theta\) with \(\frac{1}{2}(1 + \cos 2\theta)\). $$A_{\text{half}} = \frac{a^2}{2} \int_{0}^{\pi} (1 + 2\cos \theta + \frac{1}{2} + \frac{1}{2}\cos 2\theta) \, d\theta$$ $$A_{\text{half}} = \frac{a^2}{2} \int_{0}^{\pi} (\frac{3}{2} + 2\cos \theta + \frac{1}{2}\cos 2\theta) \, d\theta$$
5. Integrate and Calculate: (Skipping the final steps for brevity, but the result should be \(A = \frac{3\pi a^2}{2}\)).

Area Between Two Polar Curves

To find the area between an outer curve \(r_2 = f_2(\theta)\) and an inner curve \(r_1 = f_1(\theta)\) between \(\alpha\) and \(\beta\), we subtract the inner area from the outer area:

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

Crucial Tip: When dealing with areas or regions, always sketch the curves first. This helps identify the correct limits of integration and which function corresponds to \(r_1\) (inner) and \(r_2\) (outer).

Chapter Summary and Quick Review

You have successfully navigated the switch from rectangular thinking to angular thinking! Here are the absolute essentials:

Quick Review Box: Further Coordinate Systems (FP3)

Definitions: Polar coordinates are \((r, \theta)\) defined relative to the Pole (origin) and Initial Line (positive x-axis).

• Polar to Cartesian: \(x = r \cos \theta\), \(y = r \sin \theta\)
• Cartesian to Polar: \(r = \sqrt{x^2 + y^2}\), \(\tan \theta = y/x\) (with critical quadrant check!)
• Area Formula: $$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$$

Keep practicing those conversions and integration techniques. With practice, polar curves will start to make perfect sense!