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

The general solution for 2sin(x-pi/3)=-sqrt(2) is x=2pin+(19pi)/(12),x=2pin+(25pi)/(12)

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Popular Trigonometry >

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

Solution

2 sin (x - \frac{π}{3}) = - 2

Solution

x = 2 πn + \frac{19 π}{12}, x = 2 πn + \frac{25 π}{12}

Degrees

x = 28 5^{\circ} + 36 0^{\circ} n, x = 37 5^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x - \frac{π}{3}) = - 2

Divide both sides by

2

2 sin (x - \frac{π}{3}) = - 2

Divide both sides by

2

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

Simplify

sin (x - \frac{π}{3}) = - \frac{2}{2}

sin (x - \frac{π}{3}) = - \frac{2}{2}

General solutions for

sin (x - \frac{π}{3}) = - \frac{2}{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}

x - \frac{π}{3} = \frac{5 π}{4} + 2 πn, x - \frac{π}{3} = \frac{7 π}{4} + 2 πn

x - \frac{π}{3} = \frac{5 π}{4} + 2 πn, x - \frac{π}{3} = \frac{7 π}{4} + 2 πn

Solve

x - \frac{π}{3} = \frac{5 π}{4} + 2 πn : x = 2 πn + \frac{19 π}{12}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

x - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{4} + 2 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}{4} + 2 πn + \frac{π}{3} : 2 πn + \frac{19 π}{12}

\frac{5 π}{4} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{5 π}{4} :

multiply the denominator and numerator by

3

\frac{5 π}{4} = \frac{5 π 3}{4 \cdot 3} = \frac{15 π}{12}

= \frac{π 4}{12} + \frac{15 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + 15 π}{12}

Add similar elements:

4 π + 15 π = 19 π

= 2 πn + \frac{19 π}{12}

x = 2 πn + \frac{19 π}{12}

x = 2 πn + \frac{19 π}{12}

x = 2 πn + \frac{19 π}{12}

Solve

x - \frac{π}{3} = \frac{7 π}{4} + 2 πn : x = 2 πn + \frac{25 π}{12}

x - \frac{π}{3} = \frac{7 π}{4} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{7 π}{4} + 2 πn

Add

\frac{π}{3}

to both sides

x - \frac{π}{3} + \frac{π}{3} = \frac{7 π}{4} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} = \frac{7 π}{4} + 2 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{7 π}{4} + 2 πn + \frac{π}{3} : 2 πn + \frac{25 π}{12}

\frac{7 π}{4} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{7 π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{7 π}{4} :

multiply the denominator and numerator by

3

\frac{7 π}{4} = \frac{7 π 3}{4 \cdot 3} = \frac{21 π}{12}

= \frac{π 4}{12} + \frac{21 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + 21 π}{12}

Add similar elements:

4 π + 21 π = 25 π

= 2 πn + \frac{25 π}{12}

x = 2 πn + \frac{25 π}{12}

x = 2 πn + \frac{25 π}{12}

x = 2 πn + \frac{25 π}{12}

x = 2 πn + \frac{19 π}{12}, x = 2 πn + \frac{25 π}{12}

Graph

Popular Examples

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cos(x)= 24/25

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x-pi/3)=-sqrt(2) ?
The general solution for 2sin(x-pi/3)=-sqrt(2) is x=2pin+(19pi)/(12),x=2pin+(25pi)/(12)