# Trigonometric Equations: Problems with Solutions

By Denitsa Dimitrova (Bulgaria)
Problem 1
$\cos(2x)-7\sin x-4=0$
Problem 2
$\cos2x-5\cos x+3=0,\ \ x\in \left[0,\frac{\pi}{2}\right]$
Problem 3
$\cos 2x-5\sin x-3=0$
Problem 4
$\cos 2x+6\sin x-5=0, x\in(0, \pi]$
Problem 5
$5\sin x-2\cos^2x-1=0$

$x\in\left[0,\frac{\pi}{2}\right]$
Problem 6
$3+\cos 2x+3\sqrt{2}\cos x=0$

$x\in(0,\pi)$
Problem 7
$\cos2x+\frac{1}{\sqrt{1+\cot^2(x)}}=0$
Problem 8
$\sin^4x+\cos^4x=\frac{5}{8}, x\in [0, \pi]$
Problem 9
$\sin^3x-3\sin^2 x+3\sin x=1$
$x\in [0, \pi]$
Problem 10
$(\cos2x-1)\cot^2x=-3\sin x$
$x\in[0,\pi)$

Problem 11
$3\cos16x+8\sin^22x\cdot \cos^22x-5=0$
$x\in[0,\pi]$
Problem 12
$\sin^2x-3\sin x\cos x+2\cos^2x=0$
$x\in [0,\frac{\pi}{2}]$
Problem 13
$\sin^2x-(\sqrt{3}+1)\sin x \cos x +\sqrt{3}\cos^2x=0$
$x \in [0, \pi]$
Problem 14
$\cos x+ \sin x = \frac{\sqrt{2}}{2}$
$x\in[0,\pi]$
Problem 15
$\sin 2x-\sqrt{3}\cos 2x=-\sqrt{3}$
Problem 16
$\sin 2x-\cos x=0$
Problem 17
$\sin 3x =\sin x$
$x\in[0,2\pi]$
Problem 18
$\sin x+\sin 3x=\sin 2x+\sin 4x$
$x\in[0,\pi]$
Problem 19
$\cos x + \cos 2x+\cos 3x = 0$
Problem 20
$\sin 4x+\sin20x-\sin12x=0$
Problem 21
$\sin 3x+\cos(x-\frac{\pi}{6})=2$

Correct:
Wrong:
Unsolved problems:
Feedback   Contact email:
The text passage contains a series of trigonometric equations presented as problems, each with a specific domain for the variable \( x \). No solutions or answers are provided for these problems.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Trigonometric Equations: Problems with Solutions

By Denitsa Dimitrova (Bulgaria)
Problem 1
$\cos(2x)-7\sin x-4=0$
Problem 2
$\cos2x-5\cos x+3=0,\ \ x\in \left[0,\frac{\pi}{2}\right]$
Problem 3
$\cos 2x-5\sin x-3=0$
Problem 4
$\cos 2x+6\sin x-5=0, x\in(0, \pi]$
Problem 5
$5\sin x-2\cos^2x-1=0$

$x\in\left[0,\frac{\pi}{2}\right]$
Problem 6
$3+\cos 2x+3\sqrt{2}\cos x=0$

$x\in(0,\pi)$
Problem 7
$\cos2x+\frac{1}{\sqrt{1+\cot^2(x)}}=0$
Problem 8
$\sin^4x+\cos^4x=\frac{5}{8}, x\in [0, \pi]$
Problem 9
$\sin^3x-3\sin^2 x+3\sin x=1$
$x\in [0, \pi]$
Problem 10
$(\cos2x-1)\cot^2x=-3\sin x$
$x\in[0,\pi)$

Problem 11
$3\cos16x+8\sin^22x\cdot \cos^22x-5=0$
$x\in[0,\pi]$
Problem 12
$\sin^2x-3\sin x\cos x+2\cos^2x=0$
$x\in [0,\frac{\pi}{2}]$
Problem 13
$\sin^2x-(\sqrt{3}+1)\sin x \cos x +\sqrt{3}\cos^2x=0$
$x \in [0, \pi]$
Problem 14
$\cos x+ \sin x = \frac{\sqrt{2}}{2}$
$x\in[0,\pi]$
Problem 15
$\sin 2x-\sqrt{3}\cos 2x=-\sqrt{3}$
Problem 16
$\sin 2x-\cos x=0$
Problem 17
$\sin 3x =\sin x$
$x\in[0,2\pi]$
Problem 18
$\sin x+\sin 3x=\sin 2x+\sin 4x$
$x\in[0,\pi]$
Problem 19
$\cos x + \cos 2x+\cos 3x = 0$
Problem 20
$\sin 4x+\sin20x-\sin12x=0$
Problem 21
$\sin 3x+\cos(x-\frac{\pi}{6})=2$

Correct:
Wrong:
Unsolved problems:
Feedback   Contact email:
user
The text passage contains a series of trigonometric equations presented as problems, each with a specific domain for the variable \( x \). No solutions or answers are provided for these problems.

TYPE: MULTIPLE_QUESTIONS