Mathematics

Analysis of triangle inequality variants:
For any real numbers $x, y \in \mathbb{R}$:

• Incorrect: By reverse triangle inequality, $|x - y| \ge ||x| - |y||$.
• Correct: Applying the triangle inequality to $x$ and $(-y)$:
  $$|x - y| = |x + (-y)| \le |x| + |-y| = |x| + |y|$$
  This holds universally for all real numbers $x$ and $y$.
• Incorrect: Counterexample: $x = 1, y = 2 \implies |3| \le 1 - 2 = -1$ (false).
• Incorrect: The true relation is $|x + y| \le |x| + |y|$.

Analysis of complex modulus properties:

• Incorrect: The quotient property is an exact equality: $|z_1 / z_2| = |z_1| / |z_2|$.
• Incorrect: Denies the true equality.
• Correct: For any two complex numbers $z_1, z_2 \in \mathbb{C}$, the modulus of their product is strictly equal to the product of their individual moduli: $|z_1 z_2| = |z_1| |z_2|$.
• Incorrect: The strict inequality is false; equality always holds for multiplication.

Step-by-step conversion to polar form:
Given $z = x + iy = -4 + i 4\sqrt{3}$.

• Modulus $r$:
  $$r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8$$
• Argument $\theta$:
  $$\cos \theta = \frac{x}{r} = \frac{-4}{8} = -\frac{1}{2}$$
  $$\sin \theta = \frac{y}{r} = \frac{4\sqrt{3}}{8} = \frac{\sqrt{3}}{2}$$
  Since $\cos \theta \lt 0$ and $\sin \theta \gt 0$, the point lies in the Second Quadrant:
  $$\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$$
• Polar form:
  $$z = 8 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$$
• Incorrect: $\pi/3$ lies in the first quadrant ($4 + i4\sqrt{3}$).
• Incorrect: $\pi/6$ corresponds to argument of $4\sqrt{3} + i4$.
• Correct: Accurately gives $r = 8$ and $\theta = 2\pi/3$.
• Incorrect: $5\pi/6$ corresponds to $-4\sqrt{3} + i4$.