Question

Тригонометрическое неравенство:

Answer 1

$\mathrm{tg}^2x-1 \ \textgreater \ 0 \\\ (\mathrm{tg}x-1) (\mathrm{tg}x+1) \ \textgreater \ 0$
Замена: $\mathrm{tg}x=a$
$(a-1)(a+1)\ \textgreater \ 0 \\\ \Rightarrow \left [ {{a\ \textgreater \ 1} \atop {a\ \textless \ -1}} \right.$
Возвращаемся к исходной переменной:
$\left [ {{\mathrm{tg}x\ \textgreater \ 1} \atop {\mathrm{tg}x\ \textless \ -1}} \right. \\\ \left [ {{x\in ( \frac{ \pi }{4}+ \pi n; \ \frac{ \pi }{2}+ \pi n) } \atop x\in ( \frac{ \pi }{2}+ \pi n; \ \frac{ 3\pi }{4}+ \pi n) }} \right. \\\ \Rightarrow x\in ( \frac{ \pi }{4}+ \pi n; \ \frac{ \pi }{2}+ \pi n) \cup ( \frac{ \pi }{2}+ \pi n; \ \frac{ 3\pi }{4}+ \pi n), \ n\in Z$

Answer 2

Trigonometric Inequality

A trigonometric inequality is an inequality that contains trigonometric functions. These inequalities involve expressions like sine, cosine, tangent, cotangent, secant, or cosecant functions. They are often solved by using properties of trigonometric functions and algebraic manipulation.

Example: An example of a trigonometric inequality is: ctg(x-π/6) < 3ctg(x+π/3).

This inequality involves the cotangent function and requires solving for the values of x that satisfy the given inequality.

The solution to this inequality involves understanding the properties of the cotangent function and applying algebraic manipulations to isolate the variable x.

Would you like to know more about how to solve this specific trigonometric inequality or any other related topic?