What is the general solution for 2sin(x-pi/6)+1=0 ?

The general solution for 2sin(x-pi/6)+1=0 is x=2pin+(4pi)/3 ,x=2pin+2pi

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2sin(x-pi/6)+1=0

Solution

2 sin (x - \frac{π}{6}) + 1 = 0

Solution

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

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x - \frac{π}{6}) + 1 = 0

Move

1

to the right side

2 sin (x - \frac{π}{6}) + 1 = 0

Subtract

1

from both sides

2 sin (x - \frac{π}{6}) + 1 - 1 = 0 - 1

Simplify

2 sin (x - \frac{π}{6}) = - 1

2 sin (x - \frac{π}{6}) = - 1

Divide both sides by

2

2 sin (x - \frac{π}{6}) = - 1

Divide both sides by

2

\frac{2 sin ( x - \frac{π}{6} )}{2} = \frac{- 1}{2}

Simplify

sin (x - \frac{π}{6}) = - \frac{1}{2}

sin (x - \frac{π}{6}) = - \frac{1}{2}

General solutions for

sin (x - \frac{π}{6}) = - \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}

x - \frac{π}{6} = \frac{7 π}{6} + 2 πn, x - \frac{π}{6} = \frac{11 π}{6} + 2 πn

x - \frac{π}{6} = \frac{7 π}{6} + 2 πn, x - \frac{π}{6} = \frac{11 π}{6} + 2 πn

Solve

x - \frac{π}{6} = \frac{7 π}{6} + 2 πn : x = 2 πn + \frac{4 π}{3}

x - \frac{π}{6} = \frac{7 π}{6} + 2 πn

Move

\frac{π}{6}

to the right side

x - \frac{π}{6} = \frac{7 π}{6} + 2 πn

Add

\frac{π}{6}

to both sides

x - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{6} + 2 πn + \frac{π}{6}

Simplify

x - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{6} + 2 πn + \frac{π}{6}

Simplify

x - \frac{π}{6} + \frac{π}{6} : x

x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= x

Simplify

\frac{7 π}{6} + 2 πn + \frac{π}{6} : 2 πn + \frac{4 π}{3}

\frac{7 π}{6} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{7 π}{6}

Combine the fractions

\frac{π}{6} + \frac{7 π}{6} : \frac{4 π}{3}

Apply rule

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

= \frac{π + 7 π}{6}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{6}

Cancel the common factor:

2

= \frac{4 π}{3}

= 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

Solve

x - \frac{π}{6} = \frac{11 π}{6} + 2 πn : x = 2 πn + 2 π

x - \frac{π}{6} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{6}

to the right side

x - \frac{π}{6} = \frac{11 π}{6} + 2 πn

Add

\frac{π}{6}

to both sides

x - \frac{π}{6} + \frac{π}{6} = \frac{11 π}{6} + 2 πn + \frac{π}{6}

Simplify

x - \frac{π}{6} + \frac{π}{6} = \frac{11 π}{6} + 2 πn + \frac{π}{6}

Simplify

x - \frac{π}{6} + \frac{π}{6} : x

x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= x

Simplify

\frac{11 π}{6} + 2 πn + \frac{π}{6} : 2 πn + 2 π

\frac{11 π}{6} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{11 π}{6}

Combine the fractions

\frac{π}{6} + \frac{11 π}{6} : 2 π

Apply rule

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

= \frac{π + 11 π}{6}

Add similar elements:

π + 11 π = 12 π

= \frac{12 π}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

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

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x-pi/6)+1=0 ?
The general solution for 2sin(x-pi/6)+1=0 is x=2pin+(4pi)/3 ,x=2pin+2pi