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Popular Trigonometry >

(1-cot(x))(csc(x-pi/3)-2)<0,-pi<,x<0

Solution

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0, - π <, x < 0

Solution

\frac{π}{4} + 2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or \frac{7 π}{6} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{4 π}{3} + 2 πn < x < 2 π + 2 πn

Interval Notation

(\frac{π}{4} + 2 πn, \frac{π}{3} + 2 πn) \cup (\frac{π}{2} + 2 πn, π + 2 πn) \cup (\frac{7 π}{6} + 2 πn, \frac{5 π}{4} + 2 πn) \cup (\frac{4 π}{3} + 2 πn, 2 π + 2 πn)

Decimal

0.78539 \dots + 2 πn < x < 1.04719 \dots + 2 πn or 1.57079 \dots + 2 πn < x < 3.14159 \dots + 2 πn or 3.66519 \dots + 2 πn < x < 3.92699 \dots + 2 πn or 4.18879 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0

Periodicity of

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) : 2 π

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2)

is composed of the following functions and periods:

cot (x)

with periodicity of

π

The compound periodicity is:

= 2 π

Express with sin, cos

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

(1 - \frac{cos ( x )}{sin ( x )}) (csc (x - \frac{π}{3}) - 2) < 0

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

(1 - \frac{cos ( x )}{sin ( x )}) (\frac{1}{sin ( x - \frac{π}{3} )} - 2) < 0

(1 - \frac{cos ( x )}{sin ( x )}) (\frac{1}{sin ( x - \frac{π}{3} )} - 2) < 0

Simplify

(1 - \frac{cos ( x )}{sin ( x )}) (\frac{1}{sin ( x - \frac{π}{3} )} - 2) : \frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )}

(1 - \frac{cos ( x )}{sin ( x )}) (\frac{1}{sin ( x - \frac{π}{3} )} - 2)

Join

1 - \frac{cos ( x )}{sin ( x )} : \frac{sin ( x ) - cos ( x )}{sin ( x )}

1 - \frac{cos ( x )}{sin ( x )}

Convert element to fraction:

1 = \frac{1 sin ( x )}{sin ( x )}

= \frac{1 \cdot sin ( x )}{sin ( x )} - \frac{cos ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot sin ( x ) - cos ( x )}{sin ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x ) - cos ( x )}{sin ( x )}

= \frac{sin ( x ) - cos ( x )}{sin ( x )} (\frac{1}{sin ( x - \frac{π}{3} )} - 2)

Join

x - \frac{π}{3} : \frac{3 x - π}{3}

x - \frac{π}{3}

Convert element to fraction:

x = \frac{x 3}{3}

= \frac{x \cdot 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{x \cdot 3 - π}{3}

= \frac{sin ( x ) - cos ( x )}{sin ( x )} (\frac{1}{sin ( \frac{3 x - π}{3} )} - 2)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( sin ( x ) - cos ( x ) ) ( \frac{1}{s i n ( \frac{x \cdot 3 - π}{3} )} - 2 )}{sin ( x )}

Join

\frac{1}{sin ( \frac{x \cdot 3 - π}{3} )} - 2 : \frac{1 - 2 sin ( \frac{3 x - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

\frac{1}{sin ( \frac{x \cdot 3 - π}{3} )} - 2

Convert element to fraction:

2 = \frac{2 sin ( \frac{x 3 - π}{3} )}{sin ( \frac{x 3 - π}{3} )}

= \frac{1}{sin ( \frac{x \cdot 3 - π}{3} )} - \frac{2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

= \frac{\frac{- 2 s i n ( \frac{3 x - π}{3} ) + 1}{s i n ( \frac{3 x - π}{3} )} ( sin ( x ) - cos ( x ) )}{sin ( x )}

Multiply

(sin (x) - cos (x)) \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )} : \frac{( - 2 sin ( \frac{3 x - π}{3} ) + 1 ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} )}

(sin (x) - cos (x)) \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 1 - 2 sin ( \frac{x \cdot 3 - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} )}

= \frac{\frac{( - 2 s i n ( \frac{3 x - π}{3} ) + 1 ) ( s i n ( x ) - c o s ( x ) )}{s i n ( \frac{x \cdot 3 - π}{3} )}}{sin ( x )}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{( 1 - 2 sin ( \frac{x \cdot 3 - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} ) sin ( x )}

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} < 0

Find the zeroes and undifined points of

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )}

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} = 0

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{7 π}{6}, x = \frac{π}{4}, x = \frac{5 π}{4}

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} = 0, 0 \leq x < 2 π

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

(1 - 2 sin (\frac{3 x - π}{3})) (sin (x) - cos (x)) = 0

Solving each part separately

1 - 2 sin (\frac{3 x - π}{3}) = 0 or sin (x) - cos (x) = 0

1 - 2 sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{7 π}{6}

1 - 2 sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π

Move

1

to the right side

1 - 2 sin (\frac{3 x - π}{3}) = 0

Subtract

1

from both sides

1 - 2 sin (\frac{3 x - π}{3}) - 1 = 0 - 1

Simplify

- 2 sin (\frac{3 x - π}{3}) = - 1

- 2 sin (\frac{3 x - π}{3}) = - 1

Divide both sides by

- 2

- 2 sin (\frac{3 x - π}{3}) = - 1

Divide both sides by

- 2

\frac{- 2 sin ( \frac{3 x - π}{3} )}{- 2} = \frac{- 1}{- 2}

Simplify

sin (\frac{3 x - π}{3}) = \frac{1}{2}

sin (\frac{3 x - π}{3}) = \frac{1}{2}

General solutions for

sin (\frac{3 x - π}{3}) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn, \frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn, \frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Solve

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn : x = 2 πn + \frac{π}{3} + \frac{π}{6}

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 ( 3 x - π )}{3} : 3 x - π

\frac{3 ( 3 x - π )}{3}

Divide the numbers:

\frac{3}{3} = 1

= 3 x - π

Simplify

3 \cdot \frac{π}{6} + 3 \cdot 2 πn : \frac{π}{2} + 6 πn

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

Move

π

to the right side

3 x - π = \frac{π}{2} + 6 πn

Add

π

to both sides

3 x - π + π = \frac{π}{2} + 6 πn + π

Simplify

3 x = \frac{π}{2} + 6 πn + π

3 x = \frac{π}{2} + 6 πn + π

Divide both sides by

3

3 x = \frac{π}{2} + 6 πn + π

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3} : 2 πn + \frac{π}{3} + \frac{π}{6}

\frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{6 πn}{3} + \frac{\frac{π}{2}}{3}

\frac{6 πn}{3} = 2 πn

\frac{6 πn}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 πn

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{3} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

Solve

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn : x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 ( 3 x - π )}{3} : 3 x - π

\frac{3 ( 3 x - π )}{3}

Divide the numbers:

\frac{3}{3} = 1

= 3 x - π

Simplify

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn : \frac{5 π}{2} + 6 πn

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 π 3}{6}

Multiply the numbers:

5 \cdot 3 = 15

= \frac{15 π}{6}

Cancel the common factor:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

Move

π

to the right side

3 x - π = \frac{5 π}{2} + 6 πn

Add

π

to both sides

3 x - π + π = \frac{5 π}{2} + 6 πn + π

Simplify

3 x = \frac{5 π}{2} + 6 πn + π

3 x = \frac{5 π}{2} + 6 πn + π

Divide both sides by

3

3 x = \frac{5 π}{2} + 6 πn + π

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3} : 2 πn + \frac{π}{3} + \frac{5 π}{6}

\frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{6 πn}{3} + \frac{\frac{5 π}{2}}{3}

\frac{6 πn}{3} = 2 πn

\frac{6 πn}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 πn

\frac{\frac{5 π}{2}}{3} = \frac{5 π}{6}

\frac{\frac{5 π}{2}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 π}{6}

= \frac{π}{3} + 2 πn + \frac{5 π}{6}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}, x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{7 π}{6}

sin (x) - cos (x) = 0, 0 \leq x < 2 π : x = \frac{π}{4}, x = \frac{5 π}{4}

sin (x) - cos (x) = 0, 0 \leq x < 2 π

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 1 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{4}, x = \frac{5 π}{4}

Combine all the solutions

x = \frac{π}{2}, x = \frac{7 π}{6}, x = \frac{π}{4}, x = \frac{5 π}{4}

Find the undefined points:

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π

Find the zeros of the denominator

sin (\frac{3 x - π}{3}) sin (x) = 0

Solving each part separately

sin (\frac{3 x - π}{3}) = 0 or sin (x) = 0

sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π : x = \frac{π}{3}, x = \frac{4 π}{3}

sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π

General solutions for

sin (\frac{3 x - π}{3}) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x - π}{3} = 0 + 2 πn, \frac{3 x - π}{3} = π + 2 πn

\frac{3 x - π}{3} = 0 + 2 πn, \frac{3 x - π}{3} = π + 2 πn

Solve

\frac{3 x - π}{3} = 0 + 2 πn : x = 2 πn + \frac{π}{3}

\frac{3 x - π}{3} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x - π}{3} = 2 πn

Multiply both sides by

3

\frac{3 x - π}{3} = 2 πn

Multiply both sides by

3

\frac{3 ( 3 x - π )}{3} = 3 \cdot 2 πn

Simplify

3 x - π = 6 πn

3 x - π = 6 πn

Move

π

to the right side

3 x - π = 6 πn

Add

π

to both sides

3 x - π + π = 6 πn + π

Simplify

3 x = 6 πn + π

3 x = 6 πn + π

Divide both sides by

3

3 x = 6 πn + π

Divide both sides by

3

\frac{3 x}{3} = \frac{6 πn}{3} + \frac{π}{3}

Simplify

x = 2 πn + \frac{π}{3}

x = 2 πn + \frac{π}{3}

Solve

\frac{3 x - π}{3} = π + 2 πn : x = \frac{4 π}{3} + 2 πn

\frac{3 x - π}{3} = π + 2 πn

Multiply both sides by

3

\frac{3 x - π}{3} = π + 2 πn

Multiply both sides by

3

\frac{3 ( 3 x - π )}{3} = 3 π + 3 \cdot 2 πn

Simplify

3 x - π = 3 π + 6 πn

3 x - π = 3 π + 6 πn

Move

π

to the right side

3 x - π = 3 π + 6 πn

Add

π

to both sides

3 x - π + π = 3 π + 6 πn + π

Simplify

3 x = 4 π + 6 πn

3 x = 4 π + 6 πn

Divide both sides by

3

3 x = 4 π + 6 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{4 π}{3} + \frac{6 πn}{3}

Simplify

x = \frac{4 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn

x = 2 πn + \frac{π}{3}, x = \frac{4 π}{3} + 2 πn

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{3}, x = \frac{4 π}{3}

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Solutions for the range

0 \leq x < 2 π

x = 0, x = π

Combine all the solutions

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π

0, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}, π, \frac{7 π}{6}, \frac{5 π}{4}, \frac{4 π}{3}

Identify the intervals

0 < x < \frac{π}{4}, \frac{π}{4} < x < \frac{π}{3}, \frac{π}{3} < x < \frac{π}{2}, \frac{π}{2} < x < π, π < x < \frac{7 π}{6}, \frac{7 π}{6} < x < \frac{5 π}{4}, \frac{5 π}{4} < x < \frac{4 π}{3}, \frac{4 π}{3} < x < 2 π

Summarize in a table:

1 - 2 sin (\frac{3 x - π}{3}) sin (x) - cos (x) sin (\frac{3 x - π}{3}) sin (x) \frac{( 1 - 2 s i n ( \frac{3 x - π}{3} ) ) ( s i n ( x ) - c o s ( x ) )}{s i n ( \frac{3 x - π}{3} ) s i n ( x )} x = 0 + - - 0 Undefined 0 < x < \frac{π}{4} + - - + + x = \frac{π}{4} + 0 - + 0 \frac{π}{4} < x < \frac{π}{3} + + - + - x = \frac{π}{3} + + 0 + Undefined \frac{π}{3} < x < \frac{π}{2} + + + + + x = \frac{π}{2} 0 + + + 0 \frac{π}{2} < x < π - + + + - x = π - + + 0 Undefined π < x < \frac{7 π}{6} - + + - + x = \frac{7 π}{6} 0 + + - 0 \frac{7 π}{6} < x < \frac{5 π}{4} + + + - - x = \frac{5 π}{4} + 0 + - 0 \frac{5 π}{4} < x < \frac{4 π}{3} + - + - + x = \frac{4 π}{3} + - 0 - Undefined \frac{4 π}{3} < x < 2 π + - - - - x = 2 π + - - 0 Undefined

Identify the intervals that satisfy the required condition:

< 0

\frac{π}{4} < x < \frac{π}{3} or \frac{π}{2} < x < π or \frac{7 π}{6} < x < \frac{5 π}{4} or \frac{4 π}{3} < x < 2 π

Apply the periodicity of

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2)

\frac{π}{4} + 2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or \frac{7 π}{6} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{4 π}{3} + 2 πn < x < 2 π + 2 πn

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