" Trigonometria (2 $$ { }^{ ext {a }} $$ parte) (1.) Resolva as seguintes inequações: $$ \begin{array}{ll} ext { a) } \operatorname{sen} x \leq \frac{1}{2} & U=[0 ; 2 \pi] \ ext { b) } \operatorname{tg} x0 & U=[0 ; \pi]\end{array} $$

Question

" Trigonometria (2 $$ { }^{	ext {a }} $$ parte) (1.) Resolva as seguintes inequações: $$ \begin{array}{ll}	ext { a) } \operatorname{sen} x \leq \frac{1}{2} & U=[0 ; 2 \pi] \ 	ext { b) } \operatorname{tg} x<\sqrt{3} & U=[0 ; 2 \pi] \ 	ext { c) } \operatorname{tg}^{2} x-1>0 & U=[0 ; \pi]\end{array} $$

UpStudy AI · Accepted Answer

Let's solve the given inequalities step by step.

### a) $$ \sin x \leq \frac{1}{2} $$ for $$ U = [0; 2\pi] $$

1. The sine function is equal to $$ \frac{1}{2} $$ at specific angles within the interval $$ [0; 2\pi] $$. These angles are:
   - $$ x = \frac{\pi}{6} $$
   - $$ x = \frac{5\pi}{6} $$

2. The sine function is less than or equal to $$ \frac{1}{2} $$ in the intervals:
   - $$ [0, \frac{\pi}{6}] $$
   - $$ [\frac{5\pi}{6}, 2\pi] $$

Thus, the solution for part (a) is:
$$
x \in [0, \frac{\pi}{6}] \cup [\frac{5\pi}{6}, 2\pi]
$$

### b) $$ \tan x < \sqrt{3} $$ for $$ U = [0; 2\pi] $$

1. The tangent function is equal to $$ \sqrt{3} $$ at:
   - $$ x = \frac{\pi}{3} $$
   - $$ x = \frac{4\pi}{3} $$

2. The tangent function is less than $$ \sqrt{3} $$ in the intervals:
   - $$ [0, \frac{\pi}{3}) $$
   - $$ (\frac{\pi}{3}, \frac{4\pi}{3}) $$
   - $$ (\frac{4\pi}{3}, 2\pi) $$

Thus, the solution for part (b) is:
$$
x \in [0, \frac{\pi}{3}) \cup (\frac{\pi}{3}, \frac{4\pi}{3}) \cup (\frac{4\pi}{3}, 2\pi)
$$

### c) $$ \tan^2 x - 1 > 0 $$ for $$ U = [0; \pi] $$

1. The inequality $$ \tan^2 x - 1 > 0 $$ can be rewritten as:
   $$
   \tan^2 x > 1
   $$
   This implies:
   $$
   |\tan x| > 1
   $$

2. The tangent function is greater than 1 in the intervals:
   - $$ \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) $$

Thus, the solution for part (c) is:
$$
x \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right)
$$

### Summary of Solutions:
- a) $$ x \in [0, \frac{\pi}{6}] \cup [\frac{5\pi}{6}, 2\pi] $$
- b) $$ x \in [0, \frac{\pi}{3}) \cup (\frac{\pi}{3}, \frac{4\pi}{3}) \cup (\frac{4\pi}{3}, 2\pi) $$
- c) $$ x \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) $$
The solutions to the inequalities are: - a) $$ x $$ is between 0 and $$ \frac{\pi}{6} $$ or between $$ \frac{5\pi}{6} $$ and $$ 2\pi $$. - b) $$ x $$ is between 0 and $$ \frac{\pi}{3} $$, between $$ \frac{\pi}{3} $$ and $$ \frac{4\pi}{3} $$, or between $$ \frac{4\pi}{3} $$ and $$ 2\pi $$. - c) $$ x $$ is between $$ \frac{\pi}{4} $$ and $$ \frac{3\pi}{4} $$.

" Trigonometria (2 parte)
(1.) Resolva as seguintes inequações:

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" Trigonometria (2 parte) (1.) Resolva as seguintes inequações: