Choose arcs, not just points

An equation like $\sin x=m$ marks meeting points with a horizontal line. The inequality $\sin x>m$ chooses the arcs above this line; $\sin x

The notation $(a,b)$ excludes the ends; $[a,b]$ includes both. Strict inequalities exclude points of equality. Points where a function does not exist are never included, even with $\ge$ or $\le$.

Sine greater or less than a constant

For $-1 $$\sin x>m\iff \alpha+2k\pi

$$\sin xThe solution is the union of these ranges for all $k\in\mathbb Z$. For non-strict versions, include the extremes.

For example, $\sin x>1/2$ is equivalent to $\pi/6+2k\pi

Cosine greater or less than a constant

For $-1 $$\cos x>m\iff-\beta+2k\pi

$$\cos xFor example, $\cos x<-1/2$ gives $2\pi/3+2k\pi

Limits of sine and cosine

For $f=\sin$ or $f=\cos$, we know $-1\le f(x)\le1$.

Condition on $m$ $f(x)>m$ $f(x)
$m<-1$ All real No real
$m=-1$ All except where $f=-1$ No real
$m=1$ No real All except where $f=1$
$m>1$ No real All real

For non-strict inequalities, treat the equality explicitly: $f\ge1$ selects only maxima; $f\le-1$, minimums only; $f\ge-1$ and $f\le1$ are valid throughout the line.

Tangent: one branch at a time

If $\gamma=\arctan m\in(-\pi/2,\pi/2)$, the tangent is increasing in each branch. Like this:

$$\tan x>m\iff\gamma+k\pi

$$\tan xAsymptotes remain excluded. In non-strict versions, only the extreme $\gamma+k\pi$ can be included.

For example, $\tan x\ge1$, in $[0,2\pi)$, gives $[\pi/4,\pi/2)\cup[5\pi/4,3\pi/2)$. Periodicity allows you to transport the same analysis to any interval.

Why not multiply without looking at the sign?

In $\sin x/\cos x>m$, multiplying by $\cos x$ preserves the direction if it is positive and reverses the direction if it is negative. In zeros, the expression does not exist. Separating by branches or making a sign board avoids losing this information.

In everyday life

  • Solar panel. The useful generation hours are the solutions of $\operatorname{sen} h > m$ to the height $h$ of the Sun.
  • Accessibility. An accessible ramp needs $\operatorname{tg}\alpha \le 0{,}0833$.
  • Cameras and games. A point is in the field of view when $\cos\theta$ is greater than the cosine of the half aperture.
The accessibility norm as an inequality: $\operatorname{tg}\alpha \le 0{,}0833$.

Exercises with resolution

1. Solve $\cos x\le1/2$ into $[0,2\pi)$.

Check resolution 1

The intersections are $\pi/3$ and $5\pi/3$. The part to the left of the line includes the extremes: $[\pi/3,5\pi/3]$.

2. Solve $\tan x<0$ into $[0,2\pi)$.

Check resolution 2

The tangent is negative in quadrants II and IV: $(\pi/2,\pi)\cup(3\pi/2,2\pi)$. Zeros and asymptotes are excluded.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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