Daily math problems for June 4, 2026 — Algebra, Limits, Complex Numbers

A semigroup is a set with an associative binary operation.

$\bullet$ Convert the numerator via sum-to-product:

$$\cos 2x-\cos 3x=2\sin \frac{5x}{2}\sin \frac{x}{2}$$

$\bullet$ Apply $\sin u\sim u$ four times. Numerator:

$$2\sin \frac{5x}{2}\sin \frac{x}{2}\sim 2\cdot \frac{5x}{2}\cdot \frac{x}{2}=\frac{5x^2}{2}$$

$\bullet$ Denominator:

$$\sin 5x\cdot \sin 8x\sim 5x\cdot 8x=40x^2$$

$\bullet$ Take the ratio:

$$\frac{5x^2\mathrm{/}2}{40x^2}=\frac{1}{16}$$

$\mathrm{\mid }z-1\mathrm{\mid }=\mathrm{\mid }z-(-1)\mathrm{\mid }$ describes the set of points equidistant from $1$ and $-1$. That's the perpendicular bisector of the segment between them, i.e. the imaginary axis $\mathrm{Re}(z)=0$.