What is the general solution for 2sin(3x-pi/3)=-sqrt(3) ?

The general solution for 2sin(3x-pi/3)=-sqrt(3) is x=(2pin)/3+(5pi)/9 ,x=(2pin)/3+(2pi)/3

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2sin(3x-pi/3)=-sqrt(3)

Solution

2 sin (3 x - \frac{π}{3}) = - 3

Solution

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

Degrees

x = 10 0^{\circ} + 12 0^{\circ} n, x = 12 0^{\circ} + 12 0^{\circ} n

Solution steps

2 sin (3 x - \frac{π}{3}) = - 3

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}

Popular Examples

2cos(2x-pi/2)=sqrt(3)

2 cos (2 x - \frac{π}{2}) = 3

cos(2x)-2sin^2(x)=0

cos (2 x) - 2 sin^{2} (x) = 0

3-2cot^2(x)=-cot^2(x)

3 - 2 cot^{2} (x) = - cot^{2} (x)

2cos(t)+1=0

2 cos (t) + 1 = 0

sin(3x)= 1/(sqrt(2))

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

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(3x-pi/3)=-sqrt(3) ?
The general solution for 2sin(3x-pi/3)=-sqrt(3) is x=(2pin)/3+(5pi)/9 ,x=(2pin)/3+(2pi)/3