Tutorial 2: Complex Numbers

Complex Numbers – Tutorial 2

Cartesian & Exponential Forms

• Cartesian form: $z = x + iy$
  • Example: $z_1 = -4 + 4i$, $z_2 = -2\sqrt{3} + 2i$
• Exponential form: $z = re^{i\theta}$ where modulus $r = |z|$, argument $\theta = \arg(z)$

Modulus & Argument

• Modulus: $|z| = \sqrt{x^2 + y^2}$
• Argument: $\arg(z) = \tan^{-1}(y/x)$ with quadrant adjustments
• $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$
• $\arg(z^n) = n \arg(z)$

De Moivre's Theorem

• $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$
• $e^{i\theta} = \cos\theta + i\sin\theta$
• $e^{ix} + e^{-ix} = 2\cos x$, $e^{ix} - e^{-ix} = 2i\sin x$

Trigonometric Identities

• Expand $\cos(nx)$ and $\sin(nx)$ using De Moivre
  • Example: $\cos(4x) = 8\cos^4 x - 8\cos^2 x + 1$
  • $\sin(4x) = 4\sin x \cos x - 8\sin^3 x \cos x$
• $\cos(5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$
• $\sin(5\theta) = 16\sin^5\theta - 20\sin^3\theta + 5\sin\theta$

Roots of Unity & Cube Roots

• Cube roots: $z^{1/3} = r^{1/3} e^{i(\theta + 2k\pi)/3}$, $k = 0,1,2$
• Tenth roots of 1: $z = e^{i 2\pi k/10}$, $k = 0,...,9$
  • Real axis: $k=0,5$ (2 roots)
  • Imaginary axis: $k=2,8$ (2 roots)
  • First quadrant: $k=1,2,3,4$ (4 roots)
  • Other quadrants: $k=6,7,8,9$

Example Applications

• Simplify expressions like $(\cos 7x + i\sin 7x)(\cos 5x - i\sin 5x) = \cos 2x + i\sin 2x$
• For $z^4 = -32i$, find all $z$ with $\text{Im}(z) > 0$