Further Mathematics | Polar coordinates

Welcome to Polar Coordinates: Navigating the Mathematical World!

Hello there! If you’ve made it to Further Pure Mathematics 2 (FP2), you’re ready for some really exciting mathematics. Up until now, you have been living in the world of Cartesian coordinates, using \((x, y)\) to locate points by moving horizontally and vertically.

However, when dealing with shapes that involve rotation, such as spirals or hearts (yes, mathematical hearts!), the \((x, y)\) system becomes clumsy.

In this chapter, we switch gears and introduce Polar Coordinates. This system is much more natural for describing curves centered around a point, focusing on distance and direction. Don't worry if this seems tricky at first; we will break down the conversions and calculations step-by-step!

What you will master in this chapter:

• Defining points using distance and angle.
• Seamlessly converting between Cartesian and Polar coordinates.
• Sketching complex polar curves like cardioids and limaçons.
• Calculating the area enclosed by these fascinating curves using integration.

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

The Basics: Pole, Initial Line, and the Point P

In the Cartesian system, we measure everything from the Origin \((0, 0)\) and the positive x-axis. Polar coordinates are defined similarly, but the language changes:

1. The Pole: This is the fixed reference point, corresponding to the Origin \((0, 0)\) in the Cartesian system.

2. The Initial Line: This is the reference line, usually taken as the positive x-axis. We measure angles from this line.

3. The Point P: A point P is defined by \((r, \theta)\):

• \(r\) (The Radius): The directed distance from the Pole to the point P. Since it’s a distance, \(r\) is usually taken to be positive (\(r \ge 0\)), though sometimes negative \(r\) is defined depending on context.
• \(\theta\) (The Angle/Argument): The angle measured anticlockwise from the Initial Line to the line segment OP. \(\theta\) is measured in radians.

Analogy: Imagine a lighthouse (the Pole). To locate a ship (Point P), you need two things: how far away it is (\(r\)) and the angle of rotation from true North or East (\(\theta\)).

Important Conventions:

• Angles are usually kept in the range \(-\pi < \theta \le \pi\) or \(0 \le \theta < 2\pi\).
• Unlike Cartesian coordinates, a single point can have multiple polar representations (e.g., \((2, \pi/4)\) is the same point as \((2, 9\pi/4)\)).

Section 2: Converting Between Systems

The first essential skill in this chapter is the ability to switch between Cartesian \((x, y)\) and Polar \((r, \theta)\). We use basic trigonometry involving a right-angled triangle formed by the Pole, the point P, and the projection onto the x-axis.

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

This is the easier conversion. If you know the distance \(r\) and the angle \(\theta\), you find \(x\) and \(y\):

Formulas:
\(x = r \cos \theta\)
\(y = r \sin \theta\)

Example: Convert the polar point \((4, \pi/3)\) to Cartesian.
\(x = 4 \cos(\pi/3) = 4 (1/2) = 2\)
\(y = 4 \sin(\pi/3) = 4 (\sqrt{3}/2) = 2\sqrt{3}\)
The Cartesian point is \((2, 2\sqrt{3})\).

Conversion 2: Cartesian \((x, y)\) to Polar \((r, \theta)\)

You need to find the distance \(r\) and the angle \(\theta\).

Formulas:
\(r^2 = x^2 + y^2\) (or \(r = \sqrt{x^2 + y^2}\))
\(\tan \theta = \frac{y}{x}\)

Accessibility Tip: Finding \(\theta\) (The Quadrant Check)

Finding \(r\) is simple (always positive root), but finding the correct angle \(\theta\) requires careful attention to the quadrant of the point \((x, y)\):

Step 1: Calculate the basic reference angle, \(\alpha = \arctan |y/x|\). This angle is always acute (\(0 < \alpha < \pi/2\)).
Step 2: Determine the quadrant of the point \((x, y)\).
Step 3: Calculate \(\theta\):

• Q1 \((x>0, y>0)\): \(\theta = \alpha\)
• Q2 \((x<0, y>0)\): \(\theta = \pi - \alpha\)
• Q3 \((x<0, y<0)\): \(\theta = \pi + \alpha\) (or \(\theta = -\pi + \alpha\))
• Q4 \((x>0, y<0)\): \(\theta = 2\pi - \alpha\) (or \(\theta = -\alpha\))

Common Mistake Alert: Students often rely only on the calculator output for \(\arctan(y/x)\). If \(x\) is negative, your calculator may give an angle in Q2 or Q4, but you must manually check if it corresponds to the correct quadrant (Q2 or Q3). Always sketch the point!

Converting Equations

You might also be asked to convert entire equations from one form to another. You simply substitute the definitions:

• Example (Cartesian to Polar): Convert \(x^2 + y^2 = 9\).

  \(r^2 = 9\), so the polar equation is \(\mathbf{r = 3}\) (a circle centered at the Pole).

• Example (Polar to Cartesian): Convert \(r = 5 \sec \theta\).

  Recall \(\sec \theta = 1/\cos \theta\). So, \(r = 5/\cos \theta\), which means \(r \cos \theta = 5\). Since \(x = r \cos \theta\), the Cartesian equation is \(\mathbf{x = 5}\) (a vertical line).

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

Sketching polar curves is highly visual and often requires recognizing patterns based on the function \(f(\theta)\). We are typically interested in how \(r\) changes as \(\theta\) sweeps from \(0\) to \(2\pi\).

Step-by-Step Sketching Process:

1. Find Key Points: Calculate \(r\) for crucial values of \(\theta\) (e.g., \(0, \pi/2, \pi, 3\pi/2\), and any points where \(r\) is maximum or minimum).
2. Check for Symmetry: This saves a lot of calculation time!
3. Identify Tangents at the Pole: Determine where \(r = 0\).
4. Plot and Connect: Plot the calculated points and use symmetry to draw the smooth curve.

Symmetry Checks (Crucial Time Savers)

A curve \(r = f(\theta)\) possesses symmetry if:

• Symmetry about the Initial Line (x-axis): If replacing \(\theta\) with \(-\theta\) results in the same equation (i.e., \(f(-\theta) = f(\theta)\)).

  Example: \(r = 3 + 2\cos\theta\). Since \(\cos(-\theta) = \cos\theta\), the curve is symmetrical about the initial line.

• Symmetry about the Line \(\theta = \pi/2\) (y-axis): If replacing \(\theta\) with \(\pi - \theta\) results in the same equation (i.e., \(f(\pi - \theta) = f(\theta)\)).

  Example: \(r = 4 \sin \theta\). Since \(\sin(\pi - \theta) = \sin \theta\), the curve is symmetrical about the y-axis.

Tangents at the Pole (The FP2 Special Feature)

A curve passes through the Pole when its radial distance \(r\) is zero. The angles \(\alpha\) for which \(r = f(\alpha) = 0\) determine the equations of the tangent lines to the curve at the Pole.

The tangents at the pole are given by the lines \(\theta = \alpha\).

Example: Finding Tangents at the Pole

Consider the curve \(r = 2 + 4 \cos \theta\).

Set \(r = 0\):
\(0 = 2 + 4 \cos \theta\)
\(\cos \theta = -1/2\)

In the range \(0 \le \theta < 2\pi\), this occurs at \(\theta = 2\pi/3\) and \(\theta = 4\pi/3\).

The tangents at the Pole are the lines: \(\mathbf{\theta = 2\pi/3}\) and \(\mathbf{\theta = 4\pi/3}\).

Section 4: Area Enclosed by Polar Curves

The primary use of calculus in polar coordinates at this level is calculating the area swept out by the curve \(r = f(\theta)\) as \(\theta\) ranges from a starting angle \(\alpha\) to a finishing angle \(\beta\).

The Area Formula

The standard formula for the area of a sector-like region bounded by the curve \(r = f(\theta)\) and the radial lines \(\theta = \alpha\) and \(\theta = \beta\) is:

Area \(A\) \(= \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta\)

Did you know? This formula comes from summing up the areas of infinitely small circular sectors. The area of a standard sector is \(\frac{1}{2} r^2 \theta\); the infinitesimal version is \(\frac{1}{2} r^2 d\theta\).

Step-by-Step Area Calculation

1. Determine the Limits of Integration \(\alpha\) and \(\beta\)

The limits define the boundaries of the area you are calculating.

• For a closed loop (like a circle or a cardioid), the limits are often \(0\) to \(2\pi\), or sometimes, due to symmetry, we calculate the area of one half (e.g., \(0\) to \(\pi\)) and double the result.
• For areas that pass through the Pole, \(\alpha\) and \(\beta\) are often the angles where the curve first leaves the Pole and returns to it (i.e., where \(r=0\)).

2. Substitute \(r^2\) into the Formula

If \(r = f(\theta)\), then you must calculate \((f(\theta))^2\).

Important: If \(f(\theta)\) contains trigonometric functions (e.g., \(r = 1 + \cos \theta\)), when you square it, you often end up with terms like \(\cos^2 \theta\) or \(\sin^2 \theta\).

3. Use Identities to Simplify the Integrand

You cannot integrate \(\cos^2 \theta\) or \(\sin^2 \theta\) directly. You must use the double angle identities:

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

This transformation is absolutely essential for FP2 area calculations!

4. Perform the Integration and Apply Limits

Integrate the simplified expression and evaluate it between \(\alpha\) and \(\beta\).

Example: Calculating the Area of a Cardioid

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

1. Limits: Since it is a full, closed loop, we integrate from \(\alpha = 0\) to \(\beta = 2\pi\). (We will use symmetry and integrate from \(0\) to \(\pi\) and multiply by 2).

2. Substitute \(r^2\):
\(r^2 = a^2 (1 + \cos \theta)^2 = a^2 (1 + 2\cos \theta + \cos^2 \theta)\)

3. Simplify using Identity:
\(\cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta)\)
\(r^2 = a^2 \left( 1 + 2\cos \theta + \frac{1}{2} + \frac{1}{2}\cos 2\theta \right)\)
\(r^2 = a^2 \left( \frac{3}{2} + 2\cos \theta + \frac{1}{2}\cos 2\theta \right)\)

4. Integrate (using symmetry, \(2 \times \int_{0}^{\pi}\)):
\(A = 2 \times \frac{1}{2} \int_{0}^{\pi} r^2 d\theta\)
\(A = a^2 \int_{0}^{\pi} \left( \frac{3}{2} + 2\cos \theta + \frac{1}{2}\cos 2\theta \right) d\theta\)
\(A = a^2 \left[ \frac{3}{2}\theta + 2\sin \theta + \frac{1}{4}\sin 2\theta \right]_{0}^{\pi}\)

5. Evaluate:
At \(\theta = \pi\): \(\frac{3}{2}\pi + 2\sin(\pi) + \frac{1}{4}\sin(2\pi) = \frac{3}{2}\pi + 0 + 0\)
At \(\theta = 0\): \(0 + 0 + 0 = 0\)

Area \(A = a^2 \left( \frac{3}{2}\pi \right) = \mathbf{\frac{3}{2} \pi a^2}\).

Dealing with Inner Loops (Limaçons)

If the polar curve has an inner loop (i.e., \(r\) becomes negative for certain angles), you must treat the area of the inner loop separately. The limits for the inner loop are typically the two angles, \(\alpha_1\) and \(\alpha_2\), where \(r=0\).

If you are asked for the area *inside the outer loop but outside the inner loop*, you calculate the total area and subtract the area of the inner loop.

Summary and Final Encouragement

You have now covered the core of polar coordinates for FP2. While the formulas look different from Cartesian methods, the underlying principles of geometry and calculus remain the same. Polar coordinates provide a powerful tool for modeling rotational and spiral phenomena that are common in physics and engineering.

The skills to practice most are:
1. Quick and accurate Cartesian \(\leftrightarrow\) Polar conversions (especially checking quadrants).
2. Finding tangents at the pole (setting \(r=0\)).
3. Mastering the integration step for area, which involves using \(\cos^2 \theta\) and \(\sin^2 \theta\) identities.

Keep practicing those past paper questions, and remember: every complex curve is just a set of distances plotted against angles. You've got this!