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2sin(3x-pi/3)=-sqrt(3),0<= x<= 2pi

Solution

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

Solution

x = \frac{5 π}{9}, x = \frac{2 π}{3}, x = \frac{11 π}{9}, x = \frac{4 π}{3}, x = \frac{17 π}{9}, x = 2 π, x = 0

Degrees

x = 10 0^{\circ}, x = 12 0^{\circ}, x = 22 0^{\circ}, x = 24 0^{\circ}, x = 34 0^{\circ}, x = 36 0^{\circ}, x = 0^{\circ}

Solution steps

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

Divide both sides by

2

2 sin (3 x - \frac{π}{3}) = - 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

3 x - \frac{π}{3} = \frac{4 π}{3} + 2 πn, 3 x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

3 x - \frac{π}{3} = \frac{4 π}{3} + 2 πn, 3 x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

Solve

3 x - \frac{π}{3} = \frac{4 π}{3} + 2 πn : x = \frac{2 πn}{3} + \frac{5 π}{9}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 3 x

Simplify

\frac{4 π}{3} + 2 πn + \frac{π}{3} : 2 πn + \frac{5 π}{3}

\frac{4 π}{3} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{4 π}{3}

Combine the fractions

\frac{π}{3} + \frac{4 π}{3} : \frac{5 π}{3}

Apply rule

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

= \frac{π + 4 π}{3}

Add similar elements:

π + 4 π = 5 π

= \frac{5 π}{3}

= 2 πn + \frac{5 π}{3}

3 x = 2 πn + \frac{5 π}{3}

3 x = 2 πn + \frac{5 π}{3}

3 x = 2 πn + \frac{5 π}{3}

Divide both sides by

3

3 x = 2 πn + \frac{5 π}{3}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{5 π}{3}}{3} : \frac{2 πn}{3} + \frac{5 π}{9}

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

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{2 πn}{3} + \frac{5 π}{9}

x = \frac{2 πn}{3} + \frac{5 π}{9}

x = \frac{2 πn}{3} + \frac{5 π}{9}

x = \frac{2 πn}{3} + \frac{5 π}{9}

Solve

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

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 3 x

Simplify

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

\frac{5 π}{3} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{3}

Combine the fractions

\frac{π}{3} + \frac{5 π}{3} : 2 π

Apply rule

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

= \frac{π + 5 π}{3}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π

= 2 πn + 2 π

3 x = 2 πn + 2 π

3 x = 2 πn + 2 π

3 x = 2 πn + 2 π

Divide both sides by

3

3 x = 2 πn + 2 π

Divide both sides by

3

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

Simplify

x = \frac{2 πn}{3} + \frac{2 π}{3}

x = \frac{2 πn}{3} + \frac{2 π}{3}

x = \frac{2 πn}{3} + \frac{5 π}{9}, x = \frac{2 πn}{3} + \frac{2 π}{3}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{5 π}{9}, x = \frac{2 π}{3}, x = \frac{11 π}{9}, x = \frac{4 π}{3}, x = \frac{17 π}{9}, x = 2 π, x = 0

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