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

The general solution for 2sin(2(x-pi/3))=-sqrt(3) is x=pin+pi,x=pin+(7pi)/6

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

Solution

2 sin (2 (x - \frac{π}{3})) = - 3

Solution

x = πn + π, x = πn + \frac{7 π}{6}

Degrees

x = 18 0^{\circ} + 18 0^{\circ} n, x = 21 0^{\circ} + 18 0^{\circ} n

Solution steps

2 sin (2 (x - \frac{π}{3})) = - 3

Divide both sides by

2

2 sin (2 (x - \frac{π}{3})) = - 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

sin (2 (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}

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

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

Solve

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 ( x - \frac{π}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x - \frac{π}{3}

Simplify

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

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

\frac{\frac{4 π}{3}}{2} = \frac{2 π}{3}

\frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x - \frac{π}{3} = \frac{2 π}{3} + πn

x - \frac{π}{3} = \frac{2 π}{3} + πn

x - \frac{π}{3} = \frac{2 π}{3} + πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{2 π}{3} + πn

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

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

\frac{2 π}{3} + πn + \frac{π}{3}

Group like terms

= πn + \frac{π}{3} + \frac{2 π}{3}

Combine the fractions

\frac{π}{3} + \frac{2 π}{3} : π

Apply rule

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

= \frac{π + 2 π}{3}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= π

= πn + π

x = πn + π

x = πn + π

x = πn + π

Solve

2 (x - \frac{π}{3}) = \frac{5 π}{3} + 2 πn : x = πn + \frac{7 π}{6}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 ( x - \frac{π}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x - \frac{π}{3}

Simplify

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

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

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

\frac{\frac{5 π}{3}}{2}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x - \frac{π}{3} = \frac{5 π}{6} + πn

x - \frac{π}{3} = \frac{5 π}{6} + πn

x - \frac{π}{3} = \frac{5 π}{6} + πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{5 π}{6} + πn

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

\frac{5 π}{6} + πn + \frac{π}{3} : πn + \frac{7 π}{6}

\frac{5 π}{6} + πn + \frac{π}{3}

Group like terms

= πn + \frac{π}{3} + \frac{5 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{6} + \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 5 π}{6}

Add similar elements:

2 π + 5 π = 7 π

= πn + \frac{7 π}{6}

x = πn + \frac{7 π}{6}

x = πn + \frac{7 π}{6}

x = πn + \frac{7 π}{6}

x = πn + π, x = πn + \frac{7 π}{6}

Popular Examples

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sqrt(3)+2sin(x)=0

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(2(x-pi/3))=-sqrt(3) ?
The general solution for 2sin(2(x-pi/3))=-sqrt(3) is x=pin+pi,x=pin+(7pi)/6