Trigonometric Inequalities: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
In which quadrants is $\sin x \cdot \cos x>\tan x$ ?
Problem 2
Solve $\frac{\cos x-1}{\tan x}>0$, if $x\in \lbrack 0,2\pi ]$
Problem 3
Solve $\sin x-\frac{1}{2}>0$ given $0\leq x\leq 2\pi $
Problem 4
Solve $\cos x+\frac{1}{3}\geq 0$ given $0\leq x\leq 2\pi $
Problem 5
Solve the trigonometric inequality: $8\left\vert \tan x\right\vert -1<0$
Problem 6
If $x\in (0,2\pi ]$ , solve $\frac{\cos x}{1-\sin 2x}<0$
Problem 7
Solve $\sin \left( x-\frac{\pi }{3}\right) >\sin x$ if $0\leq x\leq 2\pi $
Problem 8
Given the inequality $p\sin x-q\cos x>\frac{r}{2}$ with solution $x\in \left( \frac{\pi }{3},\pi \right) $.

If $\left( p,q\right) $ is a point that belongs to the circumference with radius $r$ and center $\left( 0,0\right) $,

determine $\frac{p}{q}$
Problem 9
Solve: $\sin x\geq \frac{1}{2}$; given $n\in \mathbb{Z}$
Problem 10
Solve: $\cos x\geq \frac{\sqrt{2}}{2}$; $0\leq x\leq 2\pi $

Problem 11
Determine all values of $x$ such that:
$\sin (2x)>6\cos x$, given $n\in \mathbb{Z}$
Problem 12
Solve the inequality: $\tan x\geq 1$
Problem 13
For which values of $x$, ($0\leq x\leq 2\pi $) is $\sin x>\cos x$?
Problem 14
Find all values of $x$ if $x\in (0;2\pi)$ that satisfy the following trigonometric inequality.

$\sqrt{3}\cos x<1+\sin x$
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