For the cosine, the auxiliary line is vertical: the solutions for $\cos x > m$ are to the right of it, and those for $\cos x < m$, to the left.

The story

The interval notation we use in this book, with “turned” brackets to indicate an excluded extreme — $\left]a, b\right[$ — is French tradition; it was disseminated in the 20th century by the group Nicolas Bourbaki, a collective pseudonym of French mathematicians who rewrote the foundations of mathematics in a series of books. In English-speaking countries, notation with parentheses, $(a, b)$, is more common.

Sources: Bourbaki (MacTutor) · Earliest Uses of Symbols of Set Theory and Logic (Jeff Miller).

In everyday life

  • Cameras and games. A point is in the field of view of a camera of aperture $2\beta$ when the angle $\theta$ between it and the camera axis satisfies $\cos\theta > \cos\beta$; games use exactly this test to decide whether one character “sees” another.
  • Directional antennas. A directional Wi-Fi antenna gives a good signal when the deviation $\theta$ in relation to its direction is, for example, $\cos\theta > 0{,}9$, that is, $\theta < 25{,}8^\circ$.
  • Lighting. A wall is well lit by a lamp when the light reaches $\cos\theta > \frac{1}{2}$, that is, less than $60^\circ$ from the perpendicular.
The object appears in the image when $\cos\theta > \cos\beta$: the camera and games do this test all the time.

Theory

Resolution of cos x > m

ResumoRoad map

  1. Mark the point $P_2$ on the cosine axis such that $\overline{OP_2} = m$ and draw through $P_2$ the straight line $r$ perpendicular to the axis.
  2. The images of the reals $x$ such that $\cos x > m$ are at the intersection of the cycle with the demiplane to the right of $r$.
  3. Describe the appropriate ranges, starting from $A$.

Exemplocos x > √3/2

The line $u = \frac{\sqrt{3}}{2}$ cuts the cycle in the images of $\frac{\pi}{6}$ and $\frac{11\pi}{6}$; the arc on the right contains $A$:

Ative o JavaScript para ver esta figura interativa.

$$2k\pi \le x < \frac{\pi}{6} + 2k\pi \quad\text{ou}\quad \frac{11\pi}{6} + 2k\pi < x < 2\pi + 2k\pi.$$

Resolution of cos x < m

ResumoRoad map

  1. Mark $P_2$ with $\overline{OP_2} = m$ and draw the line $r$ perpendicular to the cosine axis.
  2. The images of the reals $x$ such that $\cos x < m$ are at the intersection of the cycle with the demiplane to the left of $r$.
  3. Describe the appropriate intervals.

Exemplocos x < −1/2

Ative o JavaScript para ver esta figura interativa.

$$\frac{2\pi}{3} + 2k\pi < x < \frac{4\pi}{3} + 2k\pi.$$

As the arc does not pass through $A$, an interval is sufficient.

Product inequalities and quotient inequalities

When the inequality involves a product or a quotient of expressions, we exchange the function for an auxiliary variable ($\cos x = y$, for example), we study the sign of each factor in a sign chart and only at the end do we return to the cycle.

Exemplocos 2x / cos x ≥ 0

With $y = \cos x$: $\dfrac{2y^2 - 1}{y} \ge 0$. The numerator changes sign at $\pm\frac{\sqrt{2}}{2}$ and the denominator at $0$; the signal chart shows that the quotient is $\ge 0$ to $-\frac{\sqrt{2}}{2} \le y < 0$ or $y \ge \frac{\sqrt{2}}{2}$. Each of these conditions on $\cos x$ resolves in the cycle as above. The complete solution is in exercise 304.

The exercises in this section are numbers 286 to 310 from chapter list.