Further Mathematics | Further coordinate systems

Welcome to Further Coordinate Systems: Polar Power!

Hello future mathematicians! In your journey through A Level Maths, you've relied heavily on the good old Cartesian system \((x, y)\) – the standard square grid. But what happens when the movement or shape you are describing is inherently circular or rotational?

This chapter, Further Coordinate Systems, introduces you to the concept of Polar Coordinates. This is a crucial topic in FP3 as it unlocks the ability to describe and perform calculus on shapes that are incredibly difficult (or impossible!) to handle in Cartesian form.

Don't worry if this feels like learning a whole new language; we will break down the conversions, sketching, and calculus into clear, manageable steps. Let's get started!

---

Section 1: Introduction to Polar Coordinates \((r, \theta)\)

1.1 Defining the Polar System

Instead of locating a point by moving 'across and up' (x and y), the Polar system locates a point based on distance and angle relative to a central point (the Pole or origin).

• \(r\) (Radius/Modulus): This is the straight-line distance from the Pole to the point. \(r\) is always non-negative in geometric interpretation, but can sometimes be treated as negative when sketching curves (which means plotting in the opposite direction).
• \(\theta\) (Argument/Angle): This is the angle measured counter-clockwise from the initial line (the positive x-axis). It is usually measured in radians.

Analogy: The Lighthouse

Imagine you are directing a ship using a lighthouse. You wouldn't say "go 5 miles East and 3 miles North." You'd say "go 6 miles away from the lighthouse, on a bearing of 40 degrees." Here, 6 miles is \(r\) and 40 degrees is \(\theta\).

1.2 Conversion between Cartesian \((x, y)\) and Polar \((r, \theta)\)

The relationship between the two systems is defined by basic trigonometry (SOH CAH TOA) applied to a right-angled triangle formed by the point, the origin, and the x-axis.

From Polar to Cartesian:

If you know \((r, \theta)\), you can find \((x, y)\):
Key Formulas:
1. \(x = r \cos \theta\)
2. \(y = r \sin \theta\)

From Cartesian to Polar:

If you know \((x, y)\), you can find \((r, \theta)\):
Key Formulas:
1. \(r^2 = x^2 + y^2\) (Pythagoras)
2. \(\tan \theta = \frac{y}{x}\) (Remember to check the quadrant!)

Key Takeaway for Section 1: Polar coordinates are defined by distance \(r\) and angle \(\theta\). We use basic trig identities to move seamlessly between the Cartesian and Polar worlds.

---

Section 2: Sketching Polar Curves

A polar curve is usually given by an equation of the form \(r = f(\theta)\). Unlike Cartesian sketching, where we look for intercepts and turning points, in polar sketching, we focus on how the distance from the origin changes as we sweep through angles.

2.1 Step-by-Step Sketching Process

1. Identify Key Angles: Calculate \(r\) for simple angles like \(\theta = 0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\pi\), etc.
2. Test for Symmetry: Check if the curve is symmetrical (this saves time!):
   • Symmetry about the initial line (\(x\)-axis): Does \(f(-\theta) = f(\theta)\)? If yes, it's symmetrical about the x-axis.
   • Symmetry about the Pole (origin): Does \(f(\theta + \pi) = f(\theta)\)? If yes, it's symmetrical about the Pole.
   • Symmetry about the line \(\theta = \frac{\pi}{2}\) (\(y\)-axis): Does \(f(\pi - \theta) = f(\theta)\)? If yes, it's symmetrical about the y-axis.
3. Deal with \(r=0\): Find the angles where \(r\) passes through the origin. These are important points where the curve changes direction (e.g., forming a loop).
4. Handle Negative \(r\): If your equation gives a negative value for \(r\), you plot the point in the opposite direction (add \(\pi\) to \(\theta\)).

Did You Know? Standard Polar Shapes

Some common equations you should recognize:

• \(r = a\) (A circle centered at the origin, radius \(a\)).
• \(r = a \cos \theta\) or \(r = a \sin \theta\) (A circle passing through the origin).
• \(r = a(1 \pm \cos \theta)\) or \(r = a(1 \pm \sin \theta)\) (A Cardioid or heart shape).
• \(r = a \theta\) (A Spiral, where \(r\) increases linearly with \(\theta\)).

Key Takeaway for Section 2: Sketching involves finding key values of \(r\) for specific angles \(\theta\). Symmetry tests and recognizing negative \(r\) values are essential skills.

---

Section 3: Tangents and Normals to Polar Curves

Finding the gradient \(\frac{dy}{dx}\) of a polar curve \(r = f(\theta)\) requires using the Chain Rule and treating the coordinates parametrically.

3.1 The Parametric Connection

The polar curve \(r = f(\theta)\) can be written parametrically using the angle \(\theta\) as the parameter:
\(x = r \cos \theta = f(\theta) \cos \theta\)
\(y = r \sin \theta = f(\theta) \sin \theta\)

Therefore, the gradient \(\frac{dy}{dx}\) is given by:
\[\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\]

3.2 Step-by-Step Gradient Calculation

This is where the Product Rule becomes your best friend (and your worst enemy if you forget it!). Remember, both \(r\) and the trigonometric functions depend on \(\theta\). Let \(r' = \frac{dr}{d\theta}\).

Step 1: Find \(\frac{dx}{d\theta}\) (using the Product Rule on \(x = r \cos \theta\))

\[\frac{dx}{d\theta} = \left(\frac{dr}{d\theta}\right) \cos \theta + r \left(-\sin \theta\right)\] \[\mathbf{\frac{dx}{d\theta} = r' \cos \theta - r \sin \theta}\]

Step 2: Find \(\frac{dy}{d\theta}\) (using the Product Rule on \(y = r \sin \theta\))

\[\frac{dy}{d\theta} = \left(\frac{dr}{d\theta}\right) \sin \theta + r \left(\cos \theta\right)\] \[\mathbf{\frac{dy}{d\theta} = r' \sin \theta + r \cos \theta}\]

Step 3: Combine to find \(\frac{dy}{dx}\)

\[\mathbf{\frac{dy}{dx} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta}}\]

3.3 Special Cases: Horizontal and Vertical Tangents

To find points where the tangent is:

• Horizontal: Set the numerator equal to zero: \(\frac{dy}{d\theta} = 0\).
• Vertical: Set the denominator equal to zero: \(\frac{dx}{d\theta} = 0\).

Remember, once you find the angle \(\theta\), you must substitute it back into \(r=f(\theta)\) to find the coordinates \((r, \theta)\) or \((x, y)\) of the point.

Key Takeaway for Section 3: Calculating gradients involves applying the Product Rule to the parametric forms of \(x\) and \(y\) in terms of \(\theta\). The general formula for \(\frac{dy}{dx}\) must be derived correctly.

---

Section 4: Area in Polar Coordinates

One of the main reasons we use polar coordinates is for calculating the area enclosed by rotational shapes.

4.1 The Area Formula

In Cartesian coordinates, area is found by summing tiny rectangles (\(\int y \, dx\)). In polar coordinates, we sum tiny sectors (or wedges) of a circle.

The area \(\Delta A\) of a small sector with angle \(\Delta \theta\) and radius \(r\) is approximately \(\frac{1}{2} r^2 \Delta \theta\).

By taking the limit as \(\Delta \theta \to 0\), the total area \(A\) swept out by the radius vector from angle \(\alpha\) to \(\beta\) is given by the definite integral:
\[\mathbf{A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta}\]

4.2 Applying the Formula

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

These are the starting and ending angles that define the region you are interested in.

• If finding the area enclosed by a full loop (e.g., a cardioid), you must find the angles where \(r\) is zero. These points typically define the limits (e.g., from \(\theta=0\) to \(\theta=2\pi\), or from \(\theta=0\) to \(\theta=\pi\) if using symmetry).
• If finding the area between the curve and the origin from a specific angle to another, \(\alpha\) and \(\beta\) will be given in the question.

Step 2: Express \(r^2\) in terms of \(\theta\)

If your curve is \(r = f(\theta)\), square the entire function before integrating. e.g., If \(r = 2 + \cos \theta\), then \(r^2 = (2 + \cos \theta)^2\).

Step 3: Integrate and Simplify

You will often encounter terms like \(\cos^2 \theta\) or \(\sin^2 \theta\). You must use the Double Angle Identities to integrate these terms:

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

This is a mandatory skill in FP3! Make sure you memorize these identities.

4.3 Area Bounded by Two Curves

If you need the area between two polar curves, \(r_1 = f_1(\theta)\) (the outer curve) and \(r_2 = f_2(\theta)\) (the inner curve), the area is found by subtracting the inner area from the outer area:
\[A = \frac{1}{2} \int_{\alpha}^{\beta} (r_1^2 - r_2^2) \, d\theta\]

The limits \(\alpha\) and \(\beta\) are usually the angles where the two curves intersect. You find these by setting \(f_1(\theta) = f_2(\theta)\).

Key Takeaway for Section 4: Area in polar coordinates is defined by \(A = \frac{1}{2} \int r^2 \, d\theta\). Successful integration requires expert knowledge of double angle formulae.

---

Final Review: Summary of Essential Formulas

Keep this cheat sheet handy for quick checks during practice!

Conversion:
\(x = r \cos \theta\);   \(y = r \sin \theta\)

Gradient \(\frac{dy}{dx}\):
\[\frac{dy}{dx} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta}\] (where \(r' = \frac{dr}{d\theta}\))

Area:
\[A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\]

Congratulations! You now have the tools to handle functions in a completely rotational framework. Keep practicing those product rule applications and double angle identities—they are the most tested skills in this chapter!