🔺Trigonometry Unit 10 – Polar Coordinates and Complex Numbers

Key Concepts and Definitions

• Polar coordinates represent a point's position using distance from the origin (r) and angle from the positive x-axis (θ)
• Rectangular coordinates represent a point's position using x and y values on a Cartesian plane
• Complex numbers consist of a real part and an imaginary part in the form $a + bi$, where $i = \sqrt{-1}$
• The real part of a complex number represents the value on the real axis, while the imaginary part represents the value on the imaginary axis
• Modulus (r) of a complex number is the distance from the origin to the point representing the complex number on the complex plane
• Argument (θ) of a complex number is the angle formed by the line joining the origin to the point representing the complex number and the positive real axis
• Euler's formula establishes the relationship between trigonometric functions and complex exponentials: $e^{i\theta} = \cos\theta + i\sin\theta$

Polar Coordinate System Basics

• In the polar coordinate system, points are represented by an ordered pair $(r, \theta)$
  • $r$ is the radial coordinate, representing the distance from the origin to the point
  • $\theta$ is the angular coordinate, representing the angle formed by the line joining the origin to the point and the positive x-axis
• The origin in the polar coordinate system is called the pole
• The horizontal line extending from the pole to the right is called the polar axis
• Angles in polar coordinates are typically measured in radians, but can also be measured in degrees
• Positive angles are measured counterclockwise from the polar axis, while negative angles are measured clockwise
• The radial coordinate $r$ can be positive, negative, or zero
  • If $r > 0$, the point lies on the terminal side of the angle $\theta$
  • If $r < 0$, the point lies on the terminal side of the angle $\theta + \pi$
  • If $r = 0$, the point is at the pole (origin)

Converting Between Polar and Rectangular Coordinates

• To convert from polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$, use the following formulas:
  • $x = r \cos\theta$
  • $y = r \sin\theta$
• To convert from rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$, use the following formulas:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$, with quadrant adjustments based on the signs of $x$ and $y$
• When converting from rectangular to polar coordinates, pay attention to the quadrant of the point to determine the appropriate angle $\theta$
  • Quadrant I (x > 0, y > 0): $\theta = \tan^{-1}(\frac{y}{x})$
  • Quadrant II (x < 0, y > 0): $\theta = \tan^{-1}(\frac{y}{x}) + \pi$
  • Quadrant III (x < 0, y < 0): $\theta = \tan^{-1}(\frac{y}{x}) + \pi$
  • Quadrant IV (x > 0, y < 0): $\theta = \tan^{-1}(\frac{y}{x}) + 2\pi$
• Remember that the $\tan^{-1}$ function returns angles in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$, so quadrant adjustments are necessary for points outside the first quadrant

Graphing in Polar Coordinates

• To graph an equation in polar coordinates, create a table of values for $\theta$ and calculate the corresponding $r$ values
• Plot the points $(r, \theta)$ in the polar coordinate system by measuring the angle $\theta$ from the polar axis and the distance $r$ from the pole
• Connect the plotted points with a smooth curve to create the graph of the polar equation
• Some common polar curves include:
  • Cardioid: $r = a(1 + \cos\theta)$ or $r = a(1 - \cos\theta)$
  • Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, where $n$ is a positive integer
  • Limaçon: $r = a + b\cos\theta$ or $r = a + b\sin\theta$, where $a$ and $b$ are constants
• Symmetry in polar curves:
  • If a polar equation is unchanged when $\theta$ is replaced by $-\theta$, the curve is symmetric about the polar axis
  • If a polar equation is unchanged when $\theta$ is replaced by $\theta + \pi$, the curve is symmetric about the pole
• To find the points of intersection between a polar curve and a line, substitute the polar equation of the line into the equation of the curve and solve for $\theta$

Complex Numbers: Introduction and Representation

• Complex numbers are numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit defined as $i^2 = -1$
• The real part of a complex number $a + bi$ is $a$, and the imaginary part is $b$
• Complex numbers can be represented on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis
• The complex conjugate of a complex number $a + bi$ is $a - bi$, denoted as $\overline{a + bi}$
• The modulus (absolute value) of a complex number $a + bi$ is given by $|a + bi| = \sqrt{a^2 + b^2}$
• The argument of a complex number $a + bi$ is the angle $\theta$ formed by the line joining the origin to the point $(a, b)$ and the positive real axis, given by $\theta = \tan^{-1}(\frac{b}{a})$ with quadrant adjustments
• Complex numbers can also be represented in polar form as $r(\cos\theta + i\sin\theta)$ or $re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument

Operations with Complex Numbers

• Addition and subtraction of complex numbers:
  • $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • $(a + bi) - (c + di) = (a - c) + (b - d)i$
• Multiplication of complex numbers:
  • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Division of complex numbers:
  • $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$
• Powers of $i$:
  • $i^0 = 1$
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$ (the cycle repeats)
• De Moivre's Theorem: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$
  • Useful for finding roots and powers of complex numbers in polar form

Polar Form of Complex Numbers

• A complex number $a + bi$ can be written in polar form as $r(\cos\theta + i\sin\theta)$ or $re^{i\theta}$
  • $r = \sqrt{a^2 + b^2}$ (modulus)
  • $\theta = \tan^{-1}(\frac{b}{a})$ with quadrant adjustments (argument)
• To convert from rectangular form $(a + bi)$ to polar form $r(\cos\theta + i\sin\theta)$:
  • Calculate $r = \sqrt{a^2 + b^2}$
  • Calculate $\theta = \tan^{-1}(\frac{b}{a})$ with quadrant adjustments
• To convert from polar form $r(\cos\theta + i\sin\theta)$ to rectangular form $(a + bi)$:
  • Calculate $a = r\cos\theta$
  • Calculate $b = r\sin\theta$
• Multiplication and division of complex numbers in polar form:
  • $(r_1(\cos\theta_1 + i\sin\theta_1))(r_2(\cos\theta_2 + i\sin\theta_2)) = r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$
  • $\frac{r_1(\cos\theta_1 + i\sin\theta_1)}{r_2(\cos\theta_2 + i\sin\theta_2)} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$
• Finding roots of complex numbers using polar form and De Moivre's Theorem

Applications and Real-World Examples

• Electrical engineering: Complex numbers are used to represent impedance, admittance, and other quantities in AC circuits
  • Impedance is represented as $Z = R + jX$, where $R$ is resistance, $X$ is reactance, and $j$ is the imaginary unit (electrical engineers use $j$ instead of $i$)
• Signal processing: Complex numbers are used to represent signals and their frequency components
  • Fourier transforms convert time-domain signals to frequency-domain representations using complex exponentials
• Quantum mechanics: Complex numbers are used to represent wave functions and probability amplitudes
  • The Schrödinger equation, which describes the behavior of quantum systems, involves complex-valued wave functions
• Fractals: Some fractal patterns, such as the Mandelbrot set, are generated using complex numbers
  • The Mandelbrot set is defined by the complex quadratic polynomial $f_c(z) = z^2 + c$, where $z$ and $c$ are complex numbers
• Fluid dynamics: Complex numbers are used to represent potential flow and streamlines in two-dimensional fluid flow problems
  • The complex potential function $w(z) = \phi(x, y) + i\psi(x, y)$ combines the velocity potential $\phi$ and the stream function $\psi$
• Robotics and computer graphics: Complex numbers are used to represent rotations and transformations in 2D space
  • A rotation by an angle $\theta$ can be represented by the complex number $\cos\theta + i\sin\theta$